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This booklet shows you how to apply algebraic skills in the solution of simple equations and problems. These words appear a lot in this unit. Investigate and write a brief description of their meaning here now. Equation Opposite An algebraic expression that has an equal sign ( ) in it. Balanced Formula Variable Substitute Give this a go! Q A number plus one is multiplied by three. This is equal to seven times a number minus five. What is the value of the number? Work through the book for a great way to solve this I SERIES TOPIC
0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 How does it work? Keeping things in balance Like a balanced set of scales, an equation is an algebraic expression where the left side equals the right side. Left-hand side Right-hand side LHS RHS in equations Write the equation represented by each of these balanced scales. (i) Left-hand side a a a aa + Right-hand side + a left-hand side right-hand side when balanced (ii) Left-hand side 6 n 6n nnn nnn nn Right-hand side + n and n and n + 6n n+ left-hand side right-hand side when balanced (iii) Left-hand side m and m and m + m m m mm + Right-hand side m and m and m + m+ m+ left-hand side right-hand side when balanced (iv) Write the new equation for (iii) if three ms are added to both sides. Left-hand side Right-hand side ( + ) m and ( + ) m and 5m + 6m + 5m+ 6m+ I SERIES TOPIC
0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 KEEPING THINGS IN BALANCE * KEEPING THINGS IN BALANCE * How does it work? Your Turn Keeping things in balance These scales are all in balance. Write the equation each one represents. a + b + n w.../.../0... Left-hand side Right-hand side Left-hand side Right-hand side Equation: Equation: c + xxx d + m m m Left-hand side Right-hand side Left-hand side Right-hand side Equation: Equation: Using the same balanced scales from question, write the new equation if these changes were made: a + One circle added to both sides n b + Three circles added to both sides w Left-hand side Right-hand side Left-hand side Right-hand side New equation: New equation: c + One x added to both sides d + Multiply both sides by x xx m m m Left-hand side Right-hand side Left-hand side Right-hand side New equation: New equation: I SERIES TOPIC
How does it work? Opposite operations These are pairs of mathematical operations that do the opposite to each other. The opposite to adding is subtracting. So and are opposite operations. The opposite to multiplying is dividing. Inverse Opposite So and are opposite operations. Opposite operations are also called inverse operations. It means the same thing! Use opposite operations to move clockwise (forwards) or anti-clockwise (backwards) in these puzzles. Complete these circle puzzles: (i) With numerical values only ' 5 5 + + - - - 8 + 6 ' ' Pair of opposite operations (ii) With numerical and algebraic values - 5x ' 5x + - + ' - 5 5x 5x '5 x 5 5x '5 Start here I SERIES TOPIC
How does it work? Your Turn Opposite operations Fill in all the missing gaps with values and/or operations for these circular calculations. a ' 8 8 8 6 + + 9-9 + 5 5 b 0 ' 7.5-6 7 8 c - + -.5 squared '8 ' - - and undo each other, and are another example of opposite operations - 6 - + ^- h ' ^- h d 5.5 ' cubed + 7 0-7 0 + ' - I SERIES TOPIC 5
How does it work? Your Turn Opposite operations Fill in all the missing gaps with values and/or operations for these circular calculations. a - 6 m + + 6 m - ' m + 6m - 8 - + 5m + - 5m Start here m - 8 b + 8 ' (x + 8) + 8-8 ' x x + 8 c x - y ' 6 - y ' + - + - y - (- ) (- ) ' + 8 y + 8-8 6 I SERIES TOPIC
How does it work? Your Turn Opposite operations Complete these flow charts to get the variable in each expression all by itself. a h + 6 +6 b 5r - c (c + 8) - 5 + 8 OPPOSITE OPERATIONS.../.../0... OPPOSITE OPERATIONS d + d ' + e - m - 9 5-9 ' 5 (-) f k + 5 ' + 5 g - 5 ( j + ) (-) + 5 I SERIES TOPIC 7
How does it work? Solving simple equations When asked to solve an equation, you are really being asked: What value does the variable need to be to keep the equation in balance? Simple equations like these can be solved mentally. Solve these equations: (i) a 9 + (ii) p 8 a + 9 ` a 5 p 8 p ' 8 ` p 6 Think what number plus will give 9? This number plus will equal 9 Think what number does 8 go into twice? This number divided by 8 will equal Always line up the equal signs vertically when setting out solutions Opposite operations can be used to get the variable by itself. Remember to keep the equation balanced. Solve these equations by getting the variable all by itself: (i) b 6 - (ii) x One Step equations b - + 6 + b - + 6 + b is now by itself ` b 7 x x 7 x is now by itself ` x 7 Add to both sides to keep it balanced Divide both sides by to keep it balanced Only one opposite operation is required to solve these Solutions check Always ask yourself: Does my solution feel correct? The best way to find out is by checking your answer to see. Is y - 9 the solution to the equation y- 8 y? Replacing the variable with a number is called substitution When y -9, y- 8 y becomes: -9-8 (- 9) - 7-7 ` the solution of y - 9 is correct Substitute in the number Check if the equation is balanced 8 I SERIES TOPIC
How does it work? Your Turn Solving simple equations Solve these equations mentally. a s + 5 5 b t - 5 c a 6 6 ' d - r e p + 6 5 ` s ` t ` a ` r ` p f 8 o 8 g ' i 7 h 6 + n 5 i ' t j 5. s 5 ` o ` i ` n ` t ` s 5 0 8 8 7 5 6 5 6 5 9 7 5 6 Join the dots with the answers in the same order as the questions to see what shape the variables form. 7 0 5 6 0 7 9 9 8 56 5 8 0 9 5 7 SSOLVING SIMPLE EQUATIONS Use opposite operations to solve these one step equations. a x + 7 6 b y + 5 c m - 8.../.../0... OLVING SIMPLE EQUATIONS ` x ` y ` m d a - 5 e k 7 f w ` a ` k ` w g q ' h z ' (- ) 8 i b - - 7 ` q ` z ` b j 5+ g - k - r - l 8 ' p ` g ` r ` p I SERIES TOPIC 9
How does it work? Your Turn Solutions check Check to see if these solutions are correct or incorrect. Show your working. a b Is a the solution to the equation 7 ' a 6? CORRECT INCORRECT 7 ' 6 6 6 Is k the solution to the equation k - 7 5? CORRECT INCORRECT c Is b 6 the solution to the equation - b? CORRECT INCORRECT d Is x -.5 the solution to the equation x - 8.5? CORRECT INCORRECT e Is m 6 the solution to the equation 5 ' m? CORRECT INCORRECT f Is q 8 the solution to the equation 9+ q? CORRECT INCORRECT g Is p - the solution to the equation p - - 0? CORRECT INCORRECT h Is n 5. the solution to the equation 5n + 0.5? CORRECT INCORRECT i Is d the solution to the equation 8d- 6 5d? CORRECT INCORRECT j Is w - the solution to the equation 8 - w w? CORRECT INCORRECT 0 I SERIES TOPIC
Where does it work? Two-step equations These need two different opposite operations to solve them. For simple two-step equations, usually the coefficient!. Remember that what is done to one side must be done to the other. Solve these equations by isolating the variable using inverse operations. (i) x + 7 x + - 7 - x x ' ' ` x Subtract from both sides x is still not by itself Divide both sides by Coefficient of x (ii) 5+ a 0 5-5+ a 0-5 a 5 a ' 5 ' ` a 7 Subtract 5 from both sides a is still not by itself Divide both sides by (iii) 0 - d 0-0 - d -0 - d - d '(- ) '(- ) ` d - Subtract 0 from both sides d is still not by itself Divide both sides by (-) Positive divided by a negative gives a negative answer Coefficient of d is - Take care with negative signs (iv) - k + - k + - - - k - - k '(- ) - '(- ) ` k or 75. Subtract from both sides k is still not by itself Divide both sides by (-) I SERIES TOPIC
Where does it work? Your Turn Two-step equations Use inverse operations to solve these two-step equations. a x + 5 b 6g - 7 EQUATIONS EQUATIONSTWO-SETP TWO-STEP.../.../0... x + 5-5 -5 6g - + 7 + x 6g x ' ' 6g ' ' x g c a + 9 d 5w - 8 ` a ` w e b + 8 f n - 5-8 ` b ` n 0 - m 8 g h p 0 - m -0 8-0 - m 5-6 - 9 - m ' ' m ` p i - g 8 j 5- y 6 ` g ` y k 9- q l 5 - n ` q ` n I SERIES TOPIC
Where does it work? with variables on both sides The aim is to get all the variables on one side and the numbers on the other side of the equation. Solve these equations using inverse operations. (i) 5x x+ 8 5x- x x- x+ 8 x 8 x ' 8' Subtract x from both sides Numbers and variable on different sides Get the variable x by itself ` x (ii) 6b 9b- 6b-9b 9b- 9b- - b - - b '(- ) - '(- ) Subtract 9b from both sides Numbers and variable on different sides Get the variable b by itself ` b (iii) h- 5-5h h+ 5h- 5-5h+ 5h 8h - 5 8h - 5+ 5 + 5 8h 9 Add 5h to both sides Add 5 to both sides Numbers and variable on different sides 8h ' 8 9 ' 8 ` h 7 or 8 875. Get the variable h by itself (iv) p+ 8-8p -p 8-6p - p 8-6p+ 6p - p+ 6p 8 + p 8 - - + p - 6 p - 6' p ' - p or p - Simplify by collecting like terms Add 6p to both sides Subtract from both sides Get the variable p by itself I SERIES TOPIC
Where does it work? Your Turn with variables on both sides Use inverse operations to solve these equations containing variables on both sides. a 0b 5b+ 0 b 8 - x x VARIABLES EQUATIONS WITH ON BOTH SIDES.../.../0... ` b ` x c d 8-5d d w+ 0 7w- ` d ` w e g+ g- 5 f 5m- 7 m- ` g ` m g t- + 6t h 6 - k 9-5k ` t ` k I SERIES TOPIC
Where does it work? Your Turn with variables on both sides Use inverse operations to solve these equations containing variables on both sides. a 7 - k k+ 5 b 8+ g 8- g ` k ` g c -x- x-9 d - 6 + 9w - w ` x ` w e y- + 6y 8 + 5y f 5a+ a+ 9 a -9 ` y ` a g 0 + r- r+ r - 8 h 5p- p- + p ` r ` p I SERIES TOPIC 5
Where does it work? with fractions For equations with a fraction, multiply both sides by the value in the denominator. Use inverse operations to solve these equations containing fractions. (i) 5a -5 5a -5 5a -60 Multiply both sides by the denominator () a is still not by itself numerator denominator 5a ' 5-60' 5 ` a - Divide both sides by 5 5a 5 a, so another way to solve this simple equation is to divide both sides by 5 (ii) m - 6 m - 6 m - 6 8 Multiply both sides by the denominator () m is still not by itself m - 6+ 6 8+ 6 Add 6 to both sides ` m y (iii) - 5+ 6 When only the variable term is in fraction form, we remove the other terms first. y - 5+ 5+ 6 + 5 y 6 8 y 6 6 6 ` y 8 Add 5 to both sides y is still not by itself Multiply both sides by the denominator (6) (iv) n 9 + - n + 9-9 - -9 n - n - ` n -6 Subtract 9 from both sides n is still not by itself Multiply both sides by the denominator () Positive negative negative 6 I SERIES TOPIC
Where does it work? Your Turn with fractions Solve each of the equations below containing fractions. a v 6 v 6 b 7t v v ' ' ` v ` t c u - - 6h 5 d ` u ` h x + 6 a b y x + 6 x + - x + - ` x ` y d - 6 - c 9 d - 5 8 + a ` d ` a I SERIES TOPIC 7
Where does it work? Your Turn with fractions Solve each of the equations below containing fractions. e + b - f - 05. + p - 6 EQUATIONS WITH FRACTIONS EQUATIONS WITH FRACTIONS.../.../0... ` b ` p a y 5 + y + - 5- y y b k - 0 5 ` y ` k c + h 5 6 d m 7 5 + ` h ` m e - + x -6 5 f t + 5 6 ` x ` t 8 I SERIES TOPIC
Where does it work? with parentheses This first method works best for simple equations with only one pair of parentheses. Solve these equations containing parentheses. (i) ( x + ) 6 (ii) -( h - 5) means (-) ( x + ) ' 6 ' x + x + - - ` x -( h - 5) '(- ) '(- ) h - 5 - h - 5+ 5 - + 5 h h ' ' ` h Divide both sides by Get the variable x all by itself Remove the (- ) Get the variable h all by itself You can use the distributive law to expand the parentheses first. Remember: Distributive law: a^b+ ch ab+ ac a^b- ch ab- ac Solve these equations after first expanding using the distributive law. (i) ( a- 7) a (ii) 7-6( d + ) ^a- 7h a a- a a-a- a-a - a ` a - 7-6^d + h 7-6d- 7+ -6d- + -6d '(- 6) -6d '(- 6) ` - 5 d 6 Expand the LHS Subtract a from both sides Expand the RHS Get the variable d all by itself I SERIES TOPIC 9
EQUATIONS WITH PARENTHESES ( EQUATIONS WITH PARENTHESES ( Where does it work? Your Turn with parentheses Solve these equations without using the distributive law:.../.../0... a ^a + h b 7( n - ) ^a + h' ' 6-6 - ` a ` n c -( d + ) d ( y + ) 8 ` d ` y e -( r + ) 9 f ( x + ) - ` r ` x g -( 6- k) 5 h 5 ( - w) ` k ` w 0 I SERIES TOPIC
Where does it work? Your Turn with parentheses Solve these equations after first expanding using the distributive law: a ^b - h 8 b ^w + h 8 ( b - ) 8 b - 6 8 b - 6+ 8 + b ' ' b ` w c q - ^ + h d -^x - 6h -0 ` q ` x e - 5^t + h 5 f ^m + 5h ` t ` m g - ^8+ 5ph - h ^ a + 8 h - 9 ` p ` a I SERIES TOPIC
Where does it work? Combo time: Multi-step equations These require at least two or more steps to solve. Solve these equations by isolating the variable using inverse operations. (i) 6+ d 7 6+ d 7 6+ d 8 6+ d -6 8-6 d d ' ' ` d Multiply both side by d is still not by itself Subtract 6 from both sides d is still not by itself Divide both sides by (ii) x+ 0 x 6 Multiply each numerator by the opposite denominator if both sides are fractions. x+ 0 x 6 6 ^x+ 0h x 6 ^ x+ 0h x Cross multiply the denominators Expand the left-hand side x+ 60 x x-x+ 60 x-x Subtract x from both sides 60-0x 60 '(- 0) -0x '(- 0) Divide both sides by -0 ` - x (iii) ^- mh 5^m- 6h ^ - mh 5^m- 6h 6-9m 5m- 0 6-9m-5m 5m-5m- 0 6- m - 0 6-6- m - 0-6 - m - 6 - m '(- ) - 6 '(- ) ` m 7 Expand both sides Subtract 5m from both sides Subtract 6 from both sides Divide both sides by - I SERIES TOPIC
Where does it work? Your Turn Combo time: Multi-step equations Use opposite operations to solve these multi-step equations. x - 5-9 7+ 5y 6 COMBO TIME: MULTI-STEP EQUATIONS.../.../0... COMBO TIME: MULTI-STEP EQUATIONS ` x ` y 5b+ b a- a 7 5 6 ` b ` a I SERIES TOPIC
Where does it work? Your Turn Combo time: Multi-step equations 5h h 5 + 6 ^+ qh 6^+ qh ` h ` q ^n+ h 5^ - nh -8^k- h 6^k+ h 7 8 ` n ` k I SERIES TOPIC
Where does it work? Your Turn challenge! How many different equations can you make that give the answers below? 0+ points scored in every box?.../.../0... * AWESOME * Scoring: -step equations point -step equations points With variables on both sides points or more step equations 5 points ^5a + h 5 points! a n 5 x 9 6x- 8 - x points! I SERIES TOPIC 5
What else can you do? Word problems Word problems can be changed into equations to help solve. Write matching algebraic equations for these statements: Unless you are given a variable to use, you can choose any letter to represent a number. (i) A number added to equals. Let the number be n ` + n Pick a letter to represent the number Write the equation using the chosen variable (ii) 9 subtracted from double a number leaves. Let the number be x ` x - 9 Pick a letter to represent the number Write the equation using the chosen variables The order that the numbers in a subtraction or division are written is important. (iii) Jennifer walks d metres to the shops to meet her friends. Kylie walks an extra 0 m to meet her there. Together, they walk a total of 5 m to get the shops. Write an equation to represent this. The distance walked by Jennifer d m Use the letter requested to represent Jennifer ` Kylie walks d + 0 Jennifer + Kylie 5m ` d+ d+ 0m 5m ` d + 0 m 5m Collect like terms to simplify ALWAYS finish word problems with a final statement that answers the question asked. ` The equation that represents the distance walked by Jennifer and Kylie is: d + 0m 5m (iv) Three consecutive numbers (one after the other), when added together equal 8. Let the first number be a Pick a letter to represent the first number ` a, a+ and a + are consecutive numbers ` a+ a+ + a + 8 Write the equation by adding them all together ` a + 8 Simplify For consecutive ODD or EVEN numbers, you add instead of at each step. 6 I SERIES TOPIC
What else can you do? Your Turn Word problems Match each of the statements below with the correct equation. a Seven more than a number is equal to five... b c d e f g h A number divided by seven equals five... Five minus a number equals seven... Five added to a number equals seven... The product of five and a number equals seven... The difference between a number and five is seven... Seven times a number equals five... The quotient of seven and a number equals five... 7 n 5 5- n 7 7 n 5 n + 5 7 n + 7 5 n - 5 7 n ' 7 5 5n 7 Colour in the dot next to the expressions that match each of these statements: a Fiona s ipod is m months older than the one bought by her sister months ago. The age of Fiona s ipod is given by: m m + - m m b Janet types words per minute faster than her friend (f) when chatting on a social website. The total words per minute Janet types compared with her friend is given by: - f - f f + f - c Co Tin is one third the age of his father who is x years old. Co Tin s age is given by: x x x x + d Catherine runs m metres around a running track with friends Leif and Carol. When Catherine lapped Leif, she was 00 m in front of him and 00 m in front of Carol. Which expression will give the distance ran by all three at that time? m - 600 metre m + 600 metres m - 600 metres m-00 metres e Kyle sold t tickets for a fund raising event. Marcus sold 5 less tickets. Together they sold a total of 8 tickets. Hint: Ticket sales by Marcus t - 5 5+ t 8 8 - t 5 t - 5 8 t + 5 f Jackson is cm shorter than the height of his sister (h). Together their heights are exactly 00 cm. 00 h + 00 - h h - 00 h - 00 I SERIES TOPIC 7
WORD PROBLEMSWORD PROBLEMS WORD PROBLEMS What else can you do? Your Turn Word problems Use x as the variable to write a simplified equation that represents these problems then solve. a Find three consecutive numbers that when added together equal. The three consecutive numbers are: x, x+, x + x+ x+ + x + x + x + - - x 9 x ' 9' x ` The three consecutive numbers are:, +, +,, 5 Simplified equation Solve equation for x Use solution to find the numbers b Find three consecutive ODD numbers that when added together equal 5. Hint: Three consecutive odd numbers are: x, x+, x + c The sum of three consecutive numbers is 78. Find the three numbers..if you use x -, x and x + the equation formed is simpler to solve. Try it! d The sum of four consecutive EVEN numbers is. Find the four numbers..../.../0... WORD PROBLEMS 8 I SERIES TOPIC
What else can you do? Your Turn Word problems Write equations for the following problems and then solve. a Xian thinks of a number n. After multiplying the number by 6 and then adding, the answer is 6. What number is Xian thinking of? 6n + 6 6n + - 6-6n 6n ' 6 ' 6 n Therefore Xian is thinking of the number Problem as an equation Solve equation for n Answer with a statement b Freddy thinks of a number also. After multiplying the number by, he subtracts and the result is times the original number. What number is Freddy thinking of? Let the number be x. c Kim s pet dog weighs.5 kg less than the dog living next door. Together, both dogs weigh a total.of 5 kg. How much does Kim s dog weigh? Hint: Let the weight of the dog next door equal d kg d A number plus one is multiplied by three. This is equal to seven times a number minus five. What is the value of the number? Let the number be n. Remember me? e A number minus five is multiplied by seven and divided by two. This equals the number plus two all multiplied by three. What is the number? I SERIES TOPIC 9
What else can you do? Measurement problems Simplify and solve the following: (i) Find the value of m in this triangle: m + sum 80 o 7m 5m m+ 5m+ 7m 80 5m 80 5m 5 80 o ' ' 5 ` m o o Angles in a triangle add up to 80 o Simplify sum of the angles Solve equation for m The size of each angle can be calculated by substituting the value for m back into each angle expression. ` m 6 5m 5 60 7m 7 6 o 8 o 60 o 8 o o o o 6 + 60 + 8 80 (ii) The perimeter of the triangle below is 7 units. Calculate the value of b. b b + 5 b + 5 b+ b+ 5+ b + 5 7 7b + 0 7 7b + 0-0 7-0 Perimeter sum of all the side lengths Simplify by collecting like terms Solve equation for b 7b 6 7b ' 7 6 ' 7 ` b 9 0 I SERIES TOPIC
What else can you do? Your Turn Measurement problems Write an equation and calculate the value of the variable in each of the diagrams below: a b b 7c c b c 9c Hint: Angles add to 60 o Find the value of the variable in each of the diagrams below: a Perimeter 7 units x b Perimeter 90 units PR.../.../0... S MEASUREMENT PROBLEMS MEASUREMENT PROBLEMS x + 5x - 6d 5d + d + I SERIES TOPIC
What else can you do? Formulae Formulae show how different measurements come together for special calculations. Use substitution to calculate the values represented by the following formulae. (i) The formula V IR calculates the voltage in a circuit where V is the voltage (in volts), I is the current (in amperes) and R is the resistance of the circuit (in Ohms). Calculate the voltage V if the current I is 5 amperes and the resistance R is ohms. When I 5 and R, V 5 V 00 Volts Substitute values into the formula (ii) The length of the longest side of a right-angled triangle (c) is found using c a + b where a and b are the lengths of the other shorter sides. Calculate the length of the longest side of a right-angled triangle with short sides of length a 5 cm and b cm. When a 5 cm and b cm, c c c c 5 + 5+ 69 After substitution, we are sometimes left with an equation solve. Substitute the given values into these formulae and solve the equation for the unknown variable. (i) The perimeter of a rectangle (P) is calculated using the formula P l+ b where l represents the length of the rectangle and b represents the breadth. Calculate the breadth of the rectangle with a perimeter (P) of cm and length (l) of cm. When P and l, - + b cm - + b Substitute values into the formula Solve the equation to find b b ' b ' 6 b ` The breadth of the rectangle is 6 cm Answer question with a statement (ii) The area of a triangle is found using the formula A where b represents the length of the base side and h represents the perpendicular height of the vertex opposite the base side. What is the height of a triangle with a base length of cm and area of 5 cm? When A 5 cm and b cm, 5 h cm Substitute values into the formula 5 h Solve the equation to find b 90 h 90 ' h ' 75. h ` The breadth of the rectangle is 7.5 cm Answer question with a statement bh I SERIES TOPIC
What else can you do? Your Turn Formulae FORMULAE FORMULAE Calculate the value of these formulas using the given values for each variable. a The speed (S) of an object in kilometres per hour is given by the formula S D T. D is the distance in kilometres and T the time taken in hours. Calculate S when: (i) D 0km and T hours (ii) D. 5km and T 0.5 hours FORMULAE.../.../0... 0 0 0 0 0 50 60 70 80 90 00 0 0 0 0 50 60 70 80 FORMULAE ` The speed S km/h ` The speed S km/h b To convert the temperature from degrees Fahrenheit (F) to degrees Celsius (C) the formula is: C 5 ^F-h. Calculate C when: 9 (i) F 50 o (ii) F 8.5 o ` The converted temperature o C ` The converted temperature o C Substitute the given values into these formulae and solve the equation for the unknown variable. a The area of a rectangle is calculated using the formula Alb where l and b are the length and breadth. Calculate the length l of the following rectangles with these breadths and areas: (i) A 6 cm and b cm (ii) A 5.8 cm and b.6 cm ` The length l of the rectangle cm ` The length l of the rectangle cm b ( a b) The average of two numbers (x ) is calculated by the formula x +, where a and b are the two numbers. Calculate the number a for following average values x and given number b. (i) x and b 8 (ii) x 75. and b 6. ` The number a ` The number a I SERIES TOPIC
What else can you do? Your Turn Formulae Calculate the value of these formulas using the given values for each variable. c The speed (S) of an object in kilometres per hour is given by the formula S D T. D is the distance in kilometres and T the time taken in hours. Calculate T when: (i) D 5km and S 7 km/h (ii) D 00km and S 60km/h ` The Time T taken hours ` The Time T taken hours d To convert the temperature from degrees Fahrenheit (F) to degrees Celsius (C) the formula is: C 5 ^F-h. Calculate F when: 9 o (i) C 60 (ii) C 7.5 o ` The converted temperature o F ` The converted temperature o F e h The formula for the area of a trapezium is: A ( a+ b) where h is the distance (height) between the parallel sides and a and b are the lengths of each parallel side. Calculate the height h when: (i) A cm, a and b 7 cm (ii) A 5.8, cm a 8 and b 5cm ` The height h cm ` The height h cm I SERIES TOPIC
What else can you do? Your Turn Reflection Time Reflecting on the work covered within this booklet: What useful skills have you gained by learning about equations? Write about one way you think you could apply equations to a real life situations. If you discovered or learnt about any shortcuts to help with equations or some other cool facts, jot them down here: I SERIES TOPIC 5
0 5 6 7 8 9 0 5 6 7 8 9 0 5 6 7 8 Cheat Sheet Here is a summary of the things you need to remember for Keeping things in balance An equation is an algebraic expression where the left-hand side equals the right-hand side. Left-hand side Right-hand side Opposite operations Pairs of mathematical operations that do the opposite to each other. and are opposite operations and are opposite operations Solving simple equations Solving an equation means: What value does the variable need to be to keep the equation in balance? The aim is always to get the variable by itself on one side of the equation using opposite operations. Solutions check Always ask yourself: Does my solution feel correct? The best way to find out is by checking your answer to see. with variables on both sides The aim is to get all the variables on one side and the numbers on the other side of the equation. with fractions If one fraction on either side of the equation, cross multiply the denominators. with parentheses Two methods:. Divide both sides by the number out the front first. Word problems. Use the distributive law to expand the parentheses first. Change word problems into equations to help solve. Give one of the unknown values a variable. Formulae Use substitution to calculate the value of the formula. Sometimes an equation remains after substitution. This can then be solved as usual. 6 I SERIES TOPIC
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