Linear equations. The ups and downs of equations. Equations can find the speed of a roller coaster.

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2 Linear equations The ups and downs of equations. Equations can find the speed of a roller coaster. Imagine you are a structural engineer whose job it is to design a roller coaster that will be the most eciting ride in the show. You use algebra to determine the speed of the roller coaster during its fall. The roller coaster you have designed climbs to a great height, stops briefly, then drops rapidly. As the roller coaster falls, it will accelerate under gravity gaining speed. If we ignore the effects of friction, the speed of the roller coaster depends on the time it takes to reach the ground. Its speed can be found using the equation: v = gt where v = speed in metres per second, g = acceleration due to gravity and t = the time taken for the roller coaster to fall. In seconds, a roller coaster would have a speed of 9. m/s, which is over 100 km/h, and it would have fallen over m. What a thrill! Forum What other calculations might the rollercoaster engineers need to do? What could the potential consequences be if roller-coaster engineers made errors in their calculations? If you halved the height of the rollercoaster, will the speed at the bottom also be halved? Why learn this? Equations are a shorthand way of describing something mathematically. The relationship between variables such as velocity and height from the ground can be epressed in an equation that describes the situation. We can then predict the effects of changing the different variables. A builder might use equations to find how much weight can be hung from a ceiling. A meteorologist might use equations to predict the likelihood of rain tomorrow. After completing this chapter you will be able to: write an equation in algebra understand equivalence solve equations using a variety of techniques understand the relationship between graphs and equations solve equations with the variable on both sides of the equation use substitution to check solutions use equations to solve problems. 9

3 Recall Worksheet R.1 Worksheet R. Prepare for this chapter by attempting the following questions. If you have difficulty with a question, go to Pearson Places and download the Recall Worksheet from Pearson Reader. 1 Which of the following are equations? (a) = (b) a + (c) m + = m Solve each of the following equations. (Find the value of the pronumeral that makes the statement true.) (a) a + = 1 (b) -- b = 10 (c) c = -18 (d) d = 1 (e) e = -8 (f) f = Copy and complete these flowcharts: Worksheet R. (a) (b) 10 (c) + 1 (d) + 9 Worksheet R. Worksheet R. Worksheet R. Find the value of each of the following epressions by substituting the value given in the brackets. (a) ( = 10) (b) ( + ) ( = ) (c) ( = ) What is the inverse (or opposite) operation to each of these? (a) multiply by (b) add - (c) divide by Epand each of the following to remove the brackets. (a) ( + ) (b) ( ) Key Words backtracking flowchart inverse operation solving balanced guess, check and improve point of intersection equivalent inspection solution 98 PEARSON mathematics 8

4 The language of equations.1 An equation is a mathematical sentence that contains two epressions connected by an equals sign. The equals sign tells us that the two epressions have the same value. Equations are used when some information is unknown. Unknown amounts are called variables and these are represented by letters or symbols. For eample, + = is an equation where the pronumeral represents a variable such as length. Finding the value that makes the equation a true number sentence is called solving the equation. We can solve the equation + = to obtain the solution =. When mathematical operations are written in words, we need to write the equations using mathematical symbols before we can begin the solution process. Worked Eample 1 1 Write an equation for each of the following. Use the letter in brackets to represent the variable. (a) Seven is added to a number to give a result of ten. (a) (b) A number is multiplied by five, then four is subtracted to give a result of siteen. (b) Thinking (a) Write an epression containing the variable showing the operations that have been performed on it and equate this epression to the unknown amount. (b) 1 Write an epression containing the variable showing the operations that have been performed on it and equate this epression to the unknown amount. (a) a + = 10 (b) b = 1 Simplify. b = 1 Remember that b is written as b. To check that the value obtained for a variable is the solution, we substitute the value into each side of the equation. If the left-hand side (LHS) is equal to the right-hand side (RHS), we know that we have correctly solved the equation. This is called checking by substitution. We know that = is the solution for + = because + = is a true number sentence. Linear equations 99

5 .1 Worked Eample Check by substitution whether the value given in the brackets is the solution for the equation = ( = 1). (Does it make the equation true?) Answer yes or no. Thinking + 1 Substitute the -value into the left-hand LHS = side of the equation. ( = 1) 1 + = when = 1 18 Simplify. = ---- = RHS Check whether the left-hand side of the equation equals the right-hand side. No The value given is not the solution. Answers page 0.1 Navigator The language of equations Q1 (a) (d), Q Column 1, Q (a) (d), Q Column 1, Q, Q, Q8 Q1 (c) (f), Q Column 1, Q (b) (e), Q Column, Q, Q, Q, Q8 Q1 (e) (h), Q Column, Q (c) (f), Q Column, Q, Q, Q, Q8 1 Fluency 1 Write an equation for each of the following. Use the letter in brackets to represent the variable. (a) Eight is added to a number to give a result of twelve. (a) (b) Four is subtracted from a number to give a result of siteen. (b) (c) Nine times a number gives a result of sity-three. (c) (d) The sum of eleven and a number is zero. (f ) (e) Seven is added to three times a number to give a result of ten. (u) (f) A number is multiplied by two, then seven is added to give a result of thirteen. (v) (g) Nine is added to a number, then the result is divided by seven to give si. () (h) The sum of si and a number is multiplied by eight to give a result of zero. (z) Check by substitution whether the value given in the brackets is the solution for each of the following equations. (Does it make the equation true?) Answer yes or no. (a) m + = 9 (m = ) (b) l = -9 (l = -11) (c) n = (n = 9) (d) 10 p = (p = 1) 00 PEARSON mathematics 8

6 .1 (e) q = (q = 11) (f) -- r = 9 (r = ) 8 (g) s = (s = ) (h) a + = -10 (a = -) (i) (u 8) = (u = ) (j) ( v) = 1 (v = 8) w (k) = 8 (w = 9) (l) = ( = ) (a) Which equation describes this sentence? Five is added to a number to give a result of twelve. A + 1 = n B n + 1 = C n + = 1 D n = 1 (b) Which equation describes this sentence? A number is subtracted from nineteen to give a result of eight. A b 19 = 8 B -b 19 = 8 C b 8 = 19 D 19 b = 8 (c) Which equation describes this sentence? Si is subtracted from a number to give a result of seven. A b = B b = C b = D b = (d) Which equation describes this sentence? Five is added to three times a number to give a result of twenty. A p + 0 = B + 0 = p C (p + ) = 0 D p + = 0 (e) Which equation describes this sentence? A number is subtracted from si; this result is multiplied by four to give eight. (f) A (w ) = 8 B ( w) = 8 C w = 8 D 8 w = Which equation describes this sentence? Three is subtracted from five times a number to give a result of seven. A ( )k = B (k ) = C k = D k = Write each of these equations in words. (a) m + = 9 (b) l = 9 (c) n = (d) 10 p = (e) q = (f) -- r = 9 8 (g) s = (h) a = -10 (i) (u 8) = w (j) ( v) = 1 (k) = 8 (l) = Understanding Write an equation for each of these rules, using the given variables for each of the quantities described. (a) The area (A) of a rectangle is equal to its length (l) multiplied by its width (w). (b) The area (A) of a triangle is equal to half its base (b) multiplied by its perpendicular height (h). (c) The average speed (s) of a car is equal to the distance it travels (d) divided by the time taken (t). (d) The area of a trapezium (A) is equal to half the sum of the lengths of the parallel sides (a and b) multiplied by the distance between them (h). (e) The cost in dollars (C) of a cruise is 00 times the number of nights (n) plus (f) The ta-free price (F) of an item is of its retail price (R). 11 Linear equations 01

7 .1 Reasoning A school wants to take a class of Year 8 students on an ecursion to the zoo. The bus will cost $10 and the zoo entrance is $ per person. (a) Write an equation to show how much it will cost the school to take a class of students and staff members. (b) If the zoo entrance fee is $ per person, how much will the ecursion cost? (c) Write an equation to show the total amount the students will pay for the ecursion. (d) If the students are charged $10 each for the ecursion and the staff do not pay, show that this will not cover the cost of the ecursion. (e) As the bus can carry up to passengers, it was decided that another class of students and another staff should go on the same ecursion. If the students are now to be charged $8.0 each, use equations to show that this will cover the cost of the ecursion. (f) How much money will be left over? Sarah wants to save enough money to buy a ticket to a concert. Tickets go on sale in weeks and will cost $9.00. She has saved $0 already. (a) Write an equation to show how much money she will have in weeks if she gets $ pocket money each week for doing all her chores for the net weeks. (b) If she gets $1 per week pocket money, use your equation to show that she will have enough money to buy the ticket. (c) Sarah s friend, Rhiannon, wants to go to the same concert. She gets $y pocket money each week but owes her brother $1. Write an equation to show how much money she will have in weeks time if she repays her brother. (d) If Rhiannon gets $0 per week pocket money, use your equation to find whether she will have enough money to buy a ticket. Open-ended 8 Write at least three equations that have w = as a solution, using different operations +,, and in each equation. Puzzle Lots of letters If: A + C = A B G = G B H = G FD = F A + H = E E G = F and A H represent the numbers from 0 to, find the values of A, B, C, D, E, F, G and H. 0 PEARSON mathematics 8

8 Solving linear equations. Solving an equation means finding a value for a variable that makes the equation true. There are many methods that can be used to solve a linear equation but they can be grouped into three categories: numerical, algebraic and graphical. Solving equations numerically We can solve equations numerically using methods such as inspection or guess, check and improve. Inspection is the method you used in primary school to solve simple equations: + = ; =. We simply look at the equation to find the solution. This method only works for very simple equations. We can use guess, check and improve for more complicated equations such as = 8. We guess a value for and substitute this value into the equation. If our guess is correct, the RHS will equal the LHS. If our guess is not correct, we continue to guess until we find the solution. This can be quite a lengthy process and it may be difficult to find the eact solution, which may involve fractions and negatives. Calculators that solve equations numerically, such as graphics calculators, use this method. Worked Eample Solve the following equations numerically to find. (a) = (b) = 8 Thinking (a) 1 Write the equation and identify whether this equation can be solved by inspection. (a) = If it can, write the answer. = (b) 1 Write the equation and identify whether this equation can be solved by inspection. If it can t, use guess, check and improve. Make a guess. 9 (b) = 8 Try = LHS = = = = - RHS Linear equations 0

9 . If this is not the solution, guess again. Try = Continue guessing until a solution is found. ( = made the LHS closer to the RHS than = 1, so we increase the value for ) 9 LHS = = = -- RHS Try = 9 LHS = = = ---- = 8 = RHS Write the solution. Because LHS = RHS, = is the solution. Remember that we undo addition and subtraction before multiplication and division. Solving equations algebraically To solve equations algebraically we use equivalent equations. Equivalent equations occur when a new equation can be written that leaves the value of the variable unchanged. The left-hand side of the equation remains the same as the right-hand side. To understand the process of equivalence you have previously learnt two strategies as follows. Backtracking In Year, you learnt to solve equations by backtracking using a flowchart. Flowcharts can be used to show the sequence of steps required to build an epression and then to undo it by working backwards or backtracking. For eample, to solve + =, we build the epression + using a flowchart and the necessary operations. + + If you first added, then multiplied by you would get a different equation, ( + ) =, with a different solution. As + equals, we can move backwards along the flowchart to find the -value that makes the equation true. This process is called backtracking. +? +? Remember that to undo an operation we must use the inverse (or opposite) operation. + 0 PEARSON mathematics 8 + = is the solution.

10 . When backtracking, we can undo an operation by using the inverse operation. + and are inverse operations. and are inverse operations. We undo + and before and. Balancing scales Another strategy you used to solve equations was to imagine an equation as a set of oldfashioned scales. Like the scales, an equation is balanced if its left- and right-hand sides are equal to each other or equivalent. If you perform an operation (add, subtract, multiply or divide by a value) on only one side of an equation, it becomes unbalanced, as the two sides are no longer equivalent. To balance the scales and the equations, you must always remember: Whatever is done to one side of the equation must be done to the other side. These scales show the equation + =. + = These scales show the result of removing from both sides to give =. = These scales show the result of removing half the mass from both sides of the scales to give =. = Solving equations using equivalence Backtracking and balancing the scales are really the same process. We will compare the processes to solve the equation + =. Both use inverse operations to work backwards to find the value of the unknown. Both methods use the steps subtract and then divide by. Drawing flowcharts and scales are time-consuming methods for solving equations and do not work for more comple equations. We can solve these equations more quickly by using the idea of equivalence and the use of inverse operations on both sides of the equals sign in reverse order to find the value of the variable. The following is the algebraic process used to solve the above equation. Linear equations 0

11 . + = (subtract from both sides) + = = (divide both sides by ) = -- = Always check the signs in front of the constants and the coefficients. If they are negative, we need to undo the negative as well as the numerical value. As = - +, we undo by subtracting and dividing by -. Worked Eample Check the sign of the coefficient of the variable. If it is negative, we need to undo the negative as well as the numerical part of the coefficient. Solve each of the following equations using algebra. (a) = (b) = Thinking (a) 1 (b) 1 Write the equation and identify the last operation to be performed on the given variable ( ). This is the first operation to be undone. Use the inverse operation for this operation on both sides of the equation and simplify the equation (+ ). Identify the net operation to be undone ( ) and apply the inverse operation ( ). If one side of the equation is now the variable by itself you have found the solution. State the solution. Check by substitution that your answer is the solution. Write the equation and identify the last operation to be performed on the given variable (+ ). This is the first operation to be undone. Use the inverse operation for this operation on both sides of the equation and simplify the equation ( ). (a) = + = + = = ---- = Check: LHS = = = 1 = = RHS (b) = = - = 0 PEARSON mathematics 8

12 . Identify the net operation to be undone ( -) and apply the inverse operation ( -). If one side of the equation is now the variable by itself you have found the solution. State the solution. Check by substitution that your answer is the solution = = -1 LHS = = -1 = + = = RHS Equations can be solved by performing inverse operations on both sides of the equals sign. The order in which you perform the inverse operations is important and is the opposite of the order used in the construction process. Solving equations graphically One of the important uses of graphs is to be able to find solutions to equations. If we draw a graph of a linear equation, all points on the graph will make the equation true. We can use the graph to find the y-coordinate for any given -value and the -coordinate for any given y-value. A graph can be used to evaluate an epression. If a value for is used as the -coordinate of the graph, the y-coordinate of the point gives the value of the equation for that -value. A graph can be used to solve an equation. If a value for y is used as the y-coordinate of the graph, the -coordinate of the point gives the solution of the equation. Worked Eample Use the following graph to find the value of: (a) y when = - (b) when y = y Linear equations 0

13 . Thinking (a) 1 Locate the -value you need to use along the -ais ( = -). Draw a vertical line from the -value until you reach the graph, then draw a horizontal line to the y-ais. (a) y Read off the value for y. When = -, y = -8. (b) 1 Locate the y-value you need to use along the -ais (y = 1). Draw a horizontal line from the y-value until you reach the graph and then draw a vertical line to the -ais. (b) y Read off the value for. When y = 1, =. Worked Eample Use the graph of y = + 1 to solve: (a) + 1 = 11 (b) + 1 = - Thinking (a) 1 Use the number on the RHS of the equation as the y-value. This is the y-coordinate of a point on your graph. (a) y 11 9 y = = 08 PEARSON mathematics 8

14 . Find the -coordinate of that point. This is your solution. = (b) 1 Use the number on the RHS of the equation as the y-value. This is the y-coordinate of a point on your graph. (b) y 11 9 y = = - - Find the -coordinate of that point. This is your solution. = -. Navigator Fluency Solving linear equations Q1 Column 1, Q, Q, Q, Q, Q, Q, Q8 Column 1, Q9 Column 1, Q10 (a) & (b), Q11, Q1, Q1, Q1, Q19, Q0, Q Q1 Column, Q, Q, Q, Q, Q, Q, Q8 Column, Q9 Column, Q10 (a) & (b), Q11, Q1, Q1, Q1, Q1, Q1, Q18, Q19, Q1 1 Solve each of the following equations numerically to find. (a) = 11 (b) = (c) = - (d) = -1 (e) -- = (f) -- = -11 (g) = -11 (h) + = (i) = -1 (j) = (k) = 11 (l) = - Solve each of the following equations using algebra. Q1 Column, Q, Q, Q, Q, Q, Q, Q8 Column, Q9 Column, Q10 (c) & (d), Q11, Q1, Q1, Q1, Q1, Q1, Q1, Q18, Q19, Q1 (a) a + = (b) b 8 = (c) c + 10 = 1 (d) 8 = 11 (e) 9 = (f) 1 = 9 (g) = (h) 11 = (i) = - Answers page 0 Linear equations 09

15 . Use the following graph to find the value of: y (a) y when = -1 (b) y when = 0 (c) y when = (d) when y = (e) when y = (f) when y = - (g) y when = 1 (h) y when = -0. (i) y when =. Use the graph of y = + to solve: (a) + = 8 (b) + = -1 (c) + = y 10 8 y = b b (a) Write an equation that represents the set of scales shown. (b) Write an equation to show the mass of the two boes. (Let b = mass of a bo.) (c) Write an equation to show the mass of each bo. (d) What is the solution to the equation you found in part (a)? (a) Which of the following equations is equivalent to z 10 = 0? A z = 10 B z = 0 C z = -10 D z = -0 p (b) Which of the following equations is equivalent to =? A p = 0 B p = C p = D p = 8 10 PEARSON mathematics 8

16 . (a) To obtain from + 1 you would need to: A add 1, then multiply by B divide by, then subtract 1 C subtract 1, then multiply by D subtract 1, then divide by (b) To obtain k from -k you would need to: A add, then divide by B subtract, then multiply by C add, then divide by - D subtract, then divide by - (c) To obtain m from m you would need to: A subtract, then divide by B subtract, then divide by - C add, then subtract D add, then divide by - 8 Solve each of the following using algebra. (a) d + = -10 (b) e + = - (c) f + = -1 (d) 9 + k = 1 (e) 8 + h = 18 (f) + g = (g) 8 + 9n = -1 (h) 1 + c = -1 (i) + 9k = -1 (j) r + = 10 (k) t + 8 = 1 (l) l + 11 = (m) d +. = 8.9 (n) p +.8 =. (o) 1a +. = 11. (p) g + 1 = (q) r + = (r) 8p + = Solve each of the following equations using algebra. (a) -a + = (b) -b + 11 = (c) -c = 8 (d) -d 1 = (e) -e + = (f) -f + 9 = 1 (g) g = -1 (h) 8 h = -10 (i) -9 m = (j) r = (k) 1 p = -8 (l) 9 t = -11 (m) p = (n) 8 l = 1 (o) 9 k = 19 (p). a = -.8 (q). b = -.1 (r).9 c = -. Understanding 10 Write an equation for each of these word problems and use algebra to find the unknown number. (a) A number is multiplied by, then is subtracted to give a result of. (Let the number be n.) (b) The sum of a number and seven is multiplied by three to give a result of. (Let the number be m.) (c) The sum of five and a number is divided by nine to give a result of three. (Let the number be p.) (d) A number is divided by four, then two is added to give a result of negative one. (Let the number be q.) Linear equations 11

17 . 11 Match each equation in the left column with an equivalent equation in the right column. Equation (a) + 1 = A = (b) = B = 9 (c) = 8 C = - 1 Anita s father is twice her weight plus 1 kilograms. Her father weighs 10 kg. (a) Write an equation using a to represent Anita s weight. (b) Solve the equation to find Anita s weight. 1 This is an addition pyramid. Each brick is the sum of the two bricks below it. (a) Complete the rest of the pyramid. (b) If the top brick has a value of 0, what is the value of? Reasoning Equivalent equation (d) -- = 1 D = 8 (e) = E = (f) = - F = - (g) -- = G = (h) + = H = - (i) 9 = I = 9 (j) - = 1 J = 1 1 Jeremy has a water tank in his garden. One day, his tank had only 1000 litres left in it. During the day it rained heavily and the tank filled at a rate of 00 litres per hour. Write an equation and solve it to find how many hours it took for Jeremy s tank to contain 00 L. 1 Jamal wants to buy a puppy. He has been saving all his pocket money each week for the last weeks, and his grandmother has given him $ for his birthday. He started with no money and now has $.0. Using p to represent the amount of pocket money he gets each week, write an equation and solve it to find how much pocket money Jamal receives each week PEARSON mathematics 8

18 . 1 A light aircraft needs to be careful of its weight limits. The pilot has been told that as long as the passengers on board are of average weight the limit will not be eceeded if the passengers consist of adults and children, or 1 adult and children. (a) How many children balance one adult? (b) If the pilot was to take only children on the plane, use an equation to find how many she could take. 1 Fatima drives to and from work each day for the five-day working week, and kilometres on the weekend. She manages to travel kilometres every week in her car. Use an equation to find how far it is from her home to her workplace. 18 Paul has to take litres of water with him on a boat trip. He has one 1. L container and two other containers of equal capacity. Together, the three containers will hold litres. Use an equation to find the capacity of each of the other containers. Open-ended 19 Write an equation using: (a) multiplication and addition and then solve it. (b) multiplication and subtraction 0 The scales have apples with a bunch of 0 grapes on the left and apples with a bunch of 10 grapes on the right. Claire was trying to find how many grapes had a mass equivalent to one apple. She suggests the following as possible first steps. Remove 0 grapes from each side. Take off half of everything from both sides. (a) Eplain to Claire the faults in her suggested first steps. (b) Solve the problem in two steps. 1 Write down at least three equations that are equivalent to + = 11. Jill adds a number to both sides of an equation, then divides both sides by a different number. She gets the answer: =. What might the original equation be? Algebra fruit bowl Every object in the puzzle at right is equal to a number. Find the value of each of the given objects presented in the puzzle. The numbers given are the sum of the objects in each row or column. Sometimes, only one object will appear in a row or column. That makes the puzzle easier to solve. At other times, you will have to look for relationships among the objects. Puzzle Linear equations 1

19 . Solving more comple equations The equations in this section may look more comple than others you have solved so far. However, the process of performing inverse operations on both sides is the same. To find a solution to these equations, identify the order in which the equation was constructed around the variable, using the correct inverse order of operations. The equations need to be deconstructed (undone) step by step with inverse operations in the reverse order to the way the equation was built. The number of operations that have been performed on the variable tells you how many operations need to be performed to solve the equation. Worked Eample Solve each of the following using algebra. y t (a) = (b) = - Thinking (a) 1 Write the equation and count the number of operations that have been performed on the variable. This tells us how many inverse operations need to be performed to find the solution. (Here, there are two operations, and + 9.) Perform the first inverse operation on both sides of the equation ( 9). y (a) = Two operations need to be undone = 9 y Simplify the equation. --- = - y Perform the second inverse operation --- = - on both sides of the equation ( ). Write the solution to the equation. y = -1 y --- Check by substitution that you have found the solution. y LHS = = = = = RHS 1 PEARSON mathematics 8

20 . (b) 1 Write the equation and count the number of operations that have been performed on the variable. This tells us how many inverse operations need to be performed to find the solution. (Here, there are two operations, + and.) t + (b) = - Two operations need to be undone. Perform the first inverse operation on both sides of the equation ( ). Place brackets around numerator. ( t + ) = - Simplify the equation. t + = -1 Perform the second inverse operation on both sides of the equation ( ). t + = -1 Write the solution to the equation. t = -19 Check by substitution that you have found the solution. t + LHS = = = = - = RHS Worked Eample 8 8 Solve each of the following equations using algebra. (a) (a 9) = 1 (b) ( ) = 1 Thinking (a) 1 Write the equation. (a) (a 9) = 1 Remove the brackets first by epanding using the distributive law, then count the number of operations that have been performed on the variable. This tells us how many inverse operations need to be performed to find the solution. (Here, there are two operations, 10 and.) Perform the first inverse operation on both sides of the equation (+ ). 10a = 1 10a + = 1 + Simplify the equation. 10a = 8 10a 8 Perform the second inverse operation = on both sides of the equation ( 10) Write the solution to the equation. a =.8 or -- Linear equations 1

21 . Beware! - means - so divide by - to undo. Check by substitution that you have found the solution. LHS = (a 9) = (.8 9) = (11. 9) =. = 1 = RHS (b) 1 Write the equation. (b) ( ) = 1 Remove the brackets first by epanding using the distributive law, then count the number of operations that have been performed on the variable. This tells us how many inverse operations need to be performed to find the solution. (Here, there are two operations, and + 8.) 18 = 1 Perform inverse operations until the variable is by itself on one side of the equals sign ( 18 first, - net) = = = Write the solution to the equation. = 1 Check by substitution that you have found the solution. LHS = ( ) = ( 1) = = 1 = RHS Worked Eample 9 9 Solve each of the following equations using algebra. a (a) = - (b) k = - Thinking (a) 1 Write the equation and count the number of operations that have been performed on the variable. This tells us how many inverse operations need to be performed to find the solution. (Here, there are three operations,,, and.) Multiply both sides of the equation by the number in the denominator to eliminate the fraction ( ). a (a) = - Three operations need to be undone. a = - a = -1 1 PEARSON mathematics 8

22 . (b) 1 Continue to use inverse operations until the variable is by itself on one side of the equation (+, ). Check by substitution that you have found the solution. Write the equation and count the number of operations that have been performed on the variable. This tells us how many inverse operations need to be performed to find the solution. (Here, there are three operations,, +, and.) Multiply both sides of the equation by the number in the denominator to eliminate the fraction ( ) and simplify the equation. Continue to use inverse operations until the variable is by itself on one side of the equation (, -). Check by substitution that you have found the solution. a + = -1 + a = -18 a = a = -9 a LHS = = = = = - = RHS k (b) = - k = - k = -10 k = -10 -k = -1 -k = k = LHS = k = = = = - = RHS Watch out! Here is that negative coefficient of again. - can be thought of as -1. The fraction bar means division. It also acts as though there are brackets around everything above it. For eample, = ( + ). Linear equations 1

23 . Answers page 1. Navigator Solving more comple equations Q1 Column 1, Q Column 1, Q Column 1, Q, Q Column 1, Q, Q, Q8, Q9, Q1, Q1, Q1, Q1 Q1 Column, Q Column, Q Column, Q, Q Column, Q, Q, Q8, Q9, Q11, Q1, Q1, Q1, Q1 Q1 Column, Q Column, Q Column, Q, Q Column, Q, Q, Q8, Q10, Q11, Q1, Q1, Q1, Q1, Q1 8 9 Fluency 1 Solve each of the following using algebra. (a) c f m = 10 (b) -- + = 8 (c) = 1 9 (d) -- = - (e) -- b = - (f) -- t = -9 (g) + p + r = - (h) = - (i) = -8 9 (j) y 9 t = (k) = (l) = - 1 Solve each of the following equations using algebra. (a) ( ) = 0 (b) ( + ) = -0 (c) 8( + ) = - (d) ( + 1) = -9 (e) ( + ) = -1 (f) ( ) = 1 (g) ( ) = (h) (9 ) = 0 (i) ( ) = (j) ( ) = -1 (k) ( ) = - (l) 9( ) = - Solve each of the following equations using algebra. (a) a m = -9 (b) r = - (c) = - (d) r = 8 (e) n p = 8 (f) = 8 1 (g) r = (h) f = (i) m = 8 (j) r = - (k) p = - (l) k = -9 (a) To obtain from -- + you would need to: A multiply by, then subtract B add, then multiply by C subtract, then multiply by D divide by, then add k (b) To obtain k from you would need to: A add 8, then multiply by B multiply by, then add 8 C subtract 8, then divide by D multiply by, then subtract 8 18 PEARSON mathematics 8

24 . Solve each of the following equations. a p m (a) = 11 (b) = - (c) = -18 (d) ( a) = - (e) ( a) = 11 (f) ( a) = + + (g) = (h) = 9 (i) = (j) = (k) = (l) = Match each equation (left column) with the appropriate list of inverse operations needed to solve the equation (right column). Equation Understanding Inverse operations needed to solve the equation (a) = A, then +, then (b) = B, then, then - (c) = C, then, then - (d) = D, then, then (e) = E, then, then - (f) = F, then, then (g) = G, then, then (h) = H, then +, then (i) = I, then, then (j) = J, then, then - Write the following statements as equations and then find the value of the unknown. (a) A number is multiplied by three, seven is added, and the result is divided by four to give an answer of four. What is the number? (b) A number is divided by four, one is subtracted, and the result is doubled to give an answer of si. What is the number? (c) Two is subtracted from five times a number, then the result is divided by si to give an answer of negative twelve. What is the number? (d) Seven is added to one and a half times a number to give a result of eight. What is the number? Linear equations 19

25 . 8 Margaret has put aside a certain amount of money to buy Christmas presents for five relatives. She spends $10 of the money on wrapping paper and cards, and intends to divide the remaining money equally between each relative. However, she then decides to add $0 to her budget to buy a special present for one relative. This present costs $8. (a) Write an equation an solve it to find how much money Margaret initially put aside. (b) How much money did Margaret spend altogether? 9 Amir has a job as a telemarketer to sell a particular product. He will earn two-elevenths of the total value of the product he sells each day plus an etra $0. (a) Write an equation and solve it to find the total value of the products sold in one day if he earns $10. (b) If Amir makes sales of this product, what is the selling price of the product? 10 Rimesh planted some daffodil bulbs in his garden. The following year he found that the number of bulbs had doubled. He also planted another si bulbs. The year after that he found the number of bulbs had tripled to give him 108 daffodil bulbs. Write an equation and solve it to find how many bulbs he planted initially. Reasoning For each of the following, write an appropriate equation and solve it to answer the question. 11 The sum of the digits of a two-digit number is 1. If the first digit is three times as large as the second digit, what is the number? 1 The sum of the digits of a three-digit number is 1. If the first digit is twice as large as the third digit and the second digit is one less than the third digit, what is the number? 1 Lyndall goes to the markets each Sunday to sell flowers. At the end of each market day she spends some of her earnings, which she keeps in a special market wallet. At the beginning of July, she has a certain amount of money in her wallet. On the first weekend she doubled this amount, then spent $0. On the second weekend, she doubled the money in the wallet, then spent $. On the third weekend, she trebled the money in the wallet and spent $1. At this point, Lyndall has $10 in her market wallet. How much did she have at the beginning of July? 1 Given that the area of the following rectangle is 10 cm, determine the value of. ( + 8) cm cm Open-ended 1 Write down at least three equations that can be solved by first adding, then dividing by (a) Starting with the number, perform a sequence of three different operations on it to obtain an answer of. Then, write your sequence as an equation with instead of. (b) Make up two more three-step equations. Give the equations to your friends and see whether they can solve them. 0 PEARSON mathematics 8

26 1 Caitlyn solves the equation = 8 as follows. = 8 step 1 = 8 step = step = -- step. = 1 1 step -- (a) At what step did Caitlyn make a mistake? Describe how you would correct this mistake. (b) Solve, remembering to check your answer. Magic algebra squares Magic algebra squares work just like ordinary magic squares in that the sum of each row, column and diagonal must be equal. Use these magic squares to answer the questions below. Puzzle 9 a c + a c a b a c + a b a + a 1 1 (a) Find the value for a. (b) Find the missing number. (c) Substitute this number and the value you found for a into your magic square and copy and complete your magic square with numbers. (a) Copy and complete the magic square in terms of. (b) Find the value for if the magic sum is 9. (c) Substitute the value you found for into your magic square and redo your magic square with numbers. (a) Complete the magic square in terms of c if the magic sum is c. (b) Find the value of a, b and c if the magic sum is 0, c + a = 1 and c b = 1. (c) Substitute the values you found for a, b and c into your magic square and redo your magic square with numbers. Linear equations 1