# 5 Linear Graphs and Equations

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1 Linear Graphs and Equations. Coordinates Firstl, we recap the concept of (, ) coordinates, illustrated in the following eamples. Eample On a set of coordinate aes, plot the points A (, ), B (0, ), C (, ), D (, ), E (, 0), F (, ) The -ais and the -ais cross at the origin, (0, 0). To locate the point A (, ), go units horizontall from the origin in the positive -direction and then units verticall in the positive -direction, as shown in the diagram. B (0, ) C (, ) A (, ) E (, 0) 0 D (, ) F (, )

2 . MEP Y9 Practice Book A Eample Identif the coordinates of the points A, B, C, D, E, F, G and H shown on the following grid: C B A D H 0 F G E A (, ), B (0, ), C (, ), D (, 0), E (, ), F (0, ), G (, ), H (, 0) Eample Marc has ten square tiles like this: cm Marc places all the square tiles in a row. He starts his row like this: 0 9 For each square tile he writes down the coordinates of the corner which has a.

3 The coordinates of the first corner are (, ). (a) Write down the coordinates of the net five corners which have a. (c) Look at the numbers in the coordinates. Describe two things ou notice. Marc thinks that (, ) are the coordinates of one of the corners which have a. Eplain wh he is wrong. (d) Sam has some bigger square tiles, like this: She places them net to each other in a row, like Marc's tiles. cm Write down the coordinates of the first two corners which have a. (KS/9/Ma/Levels -/P) (a) (, ), (, ), (, ), (0, ), (, ) (c) The -coordinate increases b each time; the -coordinate remains constant at. (, ) cannot be the coordinates of a corner as is an odd number and the corners which have a all have even coordinates. (d) (, ), (, ) Eercises. Write down the coordinates of the points marked on the following grid: G E F D B 0 A C H

4 . MEP Y9 Practice Book A. On a set of coordinate aes, with values from to, values from to, plot the following points: A (, ), B (, ), C (, ), D (, ), E (, ), F (0, ), G (, 0), H (, ) What can ou sa about A, B and E?. On a suitable set of coordinate aes, join the points (, 0), (0, ) and (, 0). What shape have ou made?. Three corners of a square have coordinates (, ), (, ) and (, ). Plot these points on a grid, and state the coordinates of the other corner.. Three corners of a rectangle have coordinates (, ), (, ) and (, ). Plot these points on a grid and state the coordinates of the other corner.. Two adjacent corners of a square have coordinates (, ) and (, ). (a) What is the length of a side of the square? What are the possible coordinates of the other two points?. Daniel has some parallelogram tiles. He puts them on a grid, in a continuing pattern. He numbers each tile. The diagram shows part of the pattern of tiles on the grid. 0 Daniel marks the top right corner of each tile with a. The coordinates of the corner with a on tile number are (, ). (a) What are the coordinates of the corner with a on tile number? What are the coordinates of the corner with a on tile number 0? Eplain how ou worked out our answer. (c) Daniel sas: "One tile in the pattern has a in the corner at (, )." Eplain wh Daniel is wrong.

5 (d) Daniel marks the bottom right corner of each tile with a. Cop and complete the table to show the coordinates of each corner with a. Tile Number Coordinates of the Corner with a (, ) (e) (f) Cop and complete the statement: 'Tile number has a in the corner at (...,... ).' Cop and complete the statement: 'Tile number... has a in the corner at (0, 9).' (KS/99/Ma/Tier -/P). A robot can move about on a grid. It can move North, South, East or West. It must move one step at a time. The robot starts from the point marked It takes steps. st step: West nd step: North It gets to the point marked. on the grid below. (a) The robot starts again from the point marked. It takes steps. st step: South nd step: South Cop the grid below and mark the point it gets to with a. step North step South step West step East

6 . MEP Y9 Practice Book A The robot alwas starts from the point marked. Find all the points the robot can reach in steps. Mark each point with a on the grid ou have drawn. (c) Another robot alwas starts from the point marked on this grid. step North step South step West step East It takes steps. st step: South nd step: West rd step: West It gets to the point marked. The robot starts again from the point marked. Cop and complete the table to show two more was for the robot to get to the point marked in steps. st step South West nd step West rd step West (KS/9/Ma/Tier -/P)

7 . Straight Line Graphs We look in this section at how to calculate coordinates and plot straight line graphs. We also look at the gradient and intercept of a straight line and the equation of a straight line. The gradient of a line is a measure of its steepness. The intercept of a line is the value where the line crosses the -ais. Intercept Rise Gradient = Rise Step Step The equation of a straight line is = m + c, where m = gradient and c = intercept (where the line crosses the -ais). Eample Draw the graph with equation = +. First, find the coordinates of some points on the graph. This can be done b calculating for a range of values as shown in the table. 0 9 The points can then be plotted on a set of aes and a straight line drawn through them. 9 = + 0

8 . MEP Y9 Practice Book A Eample Calculate the gradient of each of the following lines: (a) (c) (d) (a) Rise = Gradient = = Step = Rise = Gradient = = (c) Step = Rise = Gradient = = Step = (d) Rise = Gradient = = Step =

9 Eample Determine the equation of each of the following lines: (a) (a) Intecept 9 Step = Rise = Gradient = = Intercept = So m = and c =. 0 The equation is: or = m + c = + = + 9

10 . MEP Y9 Practice Book A 9 Rise = Step = 0 Intercept Gradient = = Intercept = So m = and c =. The equation is: or = m + c ( ) = + = Eercises. (a) Cop and complete the following table for =. 0 Draw the graph of =.. Draw the graphs with the equations given below, using a new set of aes for each graph. (a) = + = (c) = (d) = + (e) = (f) =. Calculate the gradient of each of the following lines, (a) - (g): (a) (c) (d) 90

11 (e) (f) (g). Write down the equations of the lines with gradients and intercepts listed below: (a) Gradient = and intercept =. Gradient = and intercept =. (c) Gradient = and intercept =. (d) Gradient = and intercept =.. Cop and complete the following table, which gives the equation, gradient and intercept for a number of straight lines. Equation Gradient Intercept = + = + = = = 0. (a) Plot the points A, B and C with coordinates: A (, ) B (, ) C (0, 0) and join them to form a triangle. Calculate the gradient of each side of the triangle. 9

12 . MEP Y9 Practice Book A. Determine the equation of each of the following lines: (a) (c) 9 (d) (e) 9 (f)

13 . (a) On a set of aes, plot the points with coordinates (, ), (, 0), (, ) and (, ) and then draw a straight line through these points. Determine the equation of the line. 9. (a) On the same aes, draw the lines with equations = + and =. Write down the coordinates of the point where the lines cross. 0. The point A has coordinates (, ), the point B has coordinates (, ) and the point C has coordinates (, 9). (a) Plot these points on a set of aes and draw straight lines through each point to form a triangle. Determine the equation of each of the lines ou have drawn.. Look at this diagram: 0 F A E B D C 0 0 (a) The line through points A and F has the equation =. What is the equation of the line through points A and B? The line through points A and D has the equation = +. What is the equation of the line through points F and E? (c) What is the equation of the line through points B and C? (KS/9/Ma/Tier -/P) 9

14 . MEP Y9 Practice Book A. Total Number of Pins (p) PINS PINS Number of Squares (s) PINS The s give the graph p = s +. The s give the graph p = s +. The s give the graph p = s +. Selma has pins. (a) Use the correct graph to find the number of squares she can pin up with pins in each square. How man squares can she pin up with pins in each square? The line through the points for p = s + climbs more steepl than the line through the points for p = s + and p = s +. Which part of the equation p = s + tells ou how steep the line is? (c) On a cop of the grid at the beginning of this question, plot three points to show the graph for pins in each square. (d) What is the equation of this graph? (KS/9/Ma/Levels -/P) 9

15 . Linear Equations In this section we consider solving linear equations, using both algebra and graphs. Eample Solve the following equations: (a) + = = (c) = (a) + = = = = (c) = (subtracting from both sides) = + (adding to both sides) = (d) = = (dividing both sides b ) = (d) = = (multipling both sides b ) = Eample Solve the following equations: (a) + = 0 (a) + = 0 = 0 = + = (c) ( + ) = (subtracting from both sides) = (dividing both sides b ) = 9

16 . MEP Y9 Practice Book A + = + = (multipling both sides b ) + = = = (subtracting from both sides) (c) ( + ) = + = (removing brackets) = = (subtracting from both sides) = (dividing both sides b ) = Eample Solve the following equations: (a) + = + = 0 (a) + = + + = (subtracting from both sides) = = (subtracting from both sides) = 0 = 0 (adding to both sides) = 0 + (adding to both sides) = = (dividing both sides b ) = 9

17 Eample Use graphs to solve the following equations: (a) = 9 + = (a) Draw the lines = and = = 9 = = The solution is given b the value on the -ais immediatel below the point where = and = 9 cross. The solution is =. Draw the lines = + and = = + = = The lines cross where =, so this is the solution of the equation. 9

18 . MEP Y9 Practice Book A Eercises. Solve the following equations: (a) + = = (c) = (d) = 0 (e) 0 = 0 (f) = (g) + 9 = (h) = (i) = (j) = 00 (k) = 9 (l) + =. Solve the following equations: (a) + = = (c) + = + (d) = 9 (e) + = (f) = (g) = (h) + (j) ( ) = (k) ( + ) = (l) ( + ) = ( ) = (i) ( ) =. (a) + = + = (c) = + (d) + = 0 (e) + = 9 (f) + = + ( ) = ( ) (h) ( + ) = (g) +. The graph = is shown: Use the graph to solve the equations: (a) = = (c) = 0 = 9

19 . Solve the equation = 9 b drawing the graphs = and = 9.. Use a graph to solve the equation =.. (a) On the same set of aes, draw the lines with equations = + and =. Use the graph to find the solution of the equation + =. Use a graph to solve the following equations: (a) = + = 9. The following graph shows the lines with equations = +, = + and = Use the graph to solve the equations: (a) + = 0 + = 0 (c) + = + 0. On the same set of aes, draw the graphs of three straight lines and use them to solve the equations: (a) = + = (c) + = 99

20 . MEP Y9 Practice Book A. Solve these equations. Show our working. (a) = = ( ) (KS/99/Ma/Tier -/P). Parallel and Perpendicular Lines In this section we consider the particular relationship between the equations of parallel lines and perpendicular lines. The ke to this is the gradient of lines that are parallel or perpendicular to each other. Eample (a) Draw the lines with equations = = + = (a) What do the three equations have in common? The following graph shows the three lines: = + = = 0 9 Note that the three lines are parallel, all with gradient. All the equations of the lines contain ''. This is because the gradient of each line is, and so the value of m in the equation = m + c is alwas. 00

21 Parallel lines will alwas have the same gradient, and so the equations of parallel lines will alwas have the same number in front of (known as the coefficient of ). For eample, the lines with equations: = = = + 0 will all be parallel (the coefficient of is in each case). Eample The equations of four lines are listed below: A = + B = + C = D = + (a) Which line is parallel to A? Which line is parallel to B? (a) C is parallel to A, because both equations contain (the coefficient of in both cases is ). D is parallel to B, because both equations contain (the coefficient of in both cases is ). Eample The graph shows two perpendicular lines, A and B: 0 B 9 A 0

22 . MEP Y9 Practice Book A (a) (c) Calculate the gradient of A and write down its equation. Calculate the gradient of B and write down its equation. Describe how the gradients of the lines are related. 0 B 9 A (a) Gradient of A = = Intercept of A = Equation of A is = Gradient of B = = Intercept of B = Equation of B is = (c) The gradients of the lines are and. So: Gradient of B = Gradient of A 0

23 If two lines A and B are perpendicular, OR Gradient of B = Gradient of A Gradient of A Gradient of B = Eample Line A has equation = +. Write down the gradient of line B that is perpendicular, and a possible equation for B. (a) Gradient of A = Gradient of B = Gradient of B = Equation of B will be = + c. This will be perpendicular to A for an value of c, so a possible equation is = +. Eercises. (a) Draw the lines with the following equations on the same set of aes: = + = + = Draw two other lines that are parallel to these lines and write down their equations. 0

24 . MEP Y9 Practice Book A. (a) Draw the line with equation =. Draw a line parallel to = that passes through the point with coordinates (0, ) (c) Determine the equation of the second line.. The equations of five lines are listed below. A = B C D E = + = + = = + (a) Which line is parallel to A? Which line is parallel to C? (c) Are there an lines parallel to B? Eplain wh.. The diagram shows the line with equation = + and two other lines, A and B, parallel to it. = + A B (a) What is the gradient of the line A? What is the equation of the line A? (c) What is the equation of the line B?. The diagram shows the line with equation = +, and three other parallel lines. What is the equation of: = + A (a) line A, line B, B (c) line C? C 0

25 . The graph shows two lines, A and B A B 9 0 (a) Calculate the gradient of the line A. What is the equation of the line A? (c) What is the equation of the line B?. The graph shows two lines, A and B. A 0 B (a) Calculate the gradient of A. Calculate the gradient of B. (c) Eplain wh the lines are perpendicular, using our answers to (a) and. 0

26 . MEP Y9 Practice Book A. The equations of five lines are given below: A B C D E = + = + = + = + = + (a) Which line is perpendicular to A? Which line is perpendicular to B? (c) Which line is not perpendicular to an of the other lines? 9. The line A joins the points with coordinates (, ) and (, ). The line B joins the points with coordinates (, ) and (, ). The line C joins the points with coordinates (, ) and (, ). (a) Calculate the gradient of each line. Which two lines are perpendicular? 0. A line has equation = +. (a) Write down the equation of lines that are parallel to = +. Write down the equation of lines that are perpendicular to = +.. The diagram shows the graph of the straight line =. (a) On a cop of the diagram, draw the graph of the straight line =. Label our line =. = (c) Write the equation of another straight line which goes through the point (0, 0). The straight line with the equation = goes through the point (, ). On our diagram, draw the graph of the straight line =. Label our line =. 0

27 (d) Write the equation of the straight line which goes through the point (0, ) and is parallel to the straight line =. (KS/9/Ma/Tier -/P). Luc was investigating straight lines and their equations. She drew the following lines. = + = = (a) = is in each equation. Write one fact this tells ou about all the lines. The lines cross the ais at (0, ), (0, 0) and (0, ). Which part of each equation helps ou see where the line crosses the ais? (c) (d) Luc decided to investigate more lines. She needed longer aes. Where will the line = 0 cross the ais? On a cop of the graph, draw another line which is parallel to =. Write the equation of our line. (KS/9/Ma/-/P) 0

28 . Simultaneous Equations Simultaneous equations consist of two or more equations that are true at the same time. Consider the following eample: Claire and Laura are sisters; we know that (i) (ii) (iii) Claire is the elder sister, their ages added together give 0 ears, the difference between their ages is ears. Let = Claire's age, in ears and = Laura's age, in ears. + = 0 = This is an eample of a pair of simultaneous equations. In this section we consider two methods of solving pairs of simultaneous equations like these. Eample Use a graph to solve the simultaneous equations: + = 0 = We can rewrite the first equation to make the subject: + = 0 = 0 For the second equation, = = + or = = Now draw the graphs = 0 and =. 0

29 = 0 (, 9) = The lines cross at the point with coordinates (, 9), so the solution of the pair of simultaneous equation is =, = 9. Note: this means that the solution to the problem presented at the start of section. is that Claire is aged and Laura is aged 9. Eample Use a graph to solve the simultaneous equations: + = = First rearrange the equations in the form =... + = = = = 9 09

30 . = = + = or = Now draw these two graphs: The lines cross at the point with coordinates (, ), so the solution is =, =. MEP Y9 Practice Book A 9 0 = (, ) = An alternative approach is to solve simultaneous equations algebraicall, as shown in the following eamples. Eample Solve the simultaneous equations: + = 9 () + = () Note that the equations have been numbered () and (). Method Substitution Method Elimination Start with equation () + = Take equation () awa from = equation (). Now replace in equation () + = 9 () Using = + = () + = 9 = () () ( ) + =9 + = 9 In equation (), replace with. = 9 = = 9 = = = = 0

31 Finall, using = gives = = So the solution is =, = Eample Solve the simultaneous equations: + = () + = () Method Substitution Method Elimination From equation () + = Subtract equation () from = equation (). Substitute this into equation () + = () ( ) = + + = () + = = () () = = + Now replace in equation () with. = + = = = Finall use = = So the solution is, =, = = So the solution is, = =, = Eample Solve the simultaneous equations: = () + = ()

32 . MEP Y9 Practice Book A Method Substitution Method Elimination From equation () + = Subtract equation () from = equation (). Substitute this into equation () + = () = = () ( ) = = () () + = = = Now replace this in equation (). = + = = 0 = + = 0 = 0 Now substitute this into = = So the solution is, = 0, = = 0 So the solution is, = 0 = 0, = Eercises. (a) Draw the lines with equations = 0 and = +. Write down the coordinates of the point where the two lines cross. (c) What is the solution of the pair of simultaneous equations, = 0 = +

33 . (a) Draw the lines with equations = and =. Determine the coordinates of the point where the two lines cross. (c) Determine the solution of the simultaneous equations, + = + =. Use a graphical method to solve the simultaneous equations, = + =. Use a graph to solve the simultaneous equations, + = 0 + =. Two numbers, and, are such that their sum is and their difference is. (a) If the numbers are and, write down a pair of simultaneous equations in and. Use a graph to solve the simultaneous equations and hence identif the two numbers.. Michelle obtains the solution =, = to a pair of simultaneous equations b drawing the following graph: What are the equations that she has solved?

34 . MEP Y9 Practice Book A. A pair of simultaneous equations are given below: (a) + = () + = ( ) Eplain wh subtracting equation () from equation () helps to solve the equations. Solve the equations.. Solve the following pairs of simultaneous equations, using algebraic methods: (a) + = + = + = + = (c) + = (d) + = + = 0 + = (e) + = (f) + = = = 9. A pair of simultaneous equations is given below: (a) + = () + = ( ) Eplain wh ou could calculate four times equation () equation () to determine one solution. Calculate the solution of this pair of equations. 0. Solve the following pairs of simultaneous equations, using an algebraic method: (a) + = + 9 = + = + = (c) + = (d) + = 0 = + = 9 (e) = (f) = + = = 0

35 . Look at this graph: 0 A B 0 0 (a) Show that the equation of line A is + =. Write the equation of line B. (c) On a cop of the graph, draw the line whose equation is = +. Label our line C. (d) Solve these simultaneous equations: Show our working. = + = +. Look at this octagon: (a) The line through A and H has the equation = 0. What is the equation of the line through F and G? Cop the following statement, adding in the missing words to make it correct: + = is the equation of the line through... and... D E 0 C F (KS/99/Ma/Tier -/P) 0 0 B 0 0 G A H

36 . MEP Y9 Practice Book A (c) (d) The octagon has four lines of smmetr. One of the lines of smmetr has the equation =. On a cop of the diagram, draw and label the line =. The octagon has three other lines of smmetr. Write the equation of one of these three other lines of smmetr, (e) The line through D and B has the equation = +. The line through G and H has the equation = +. D C B A E F G H Solve the simultaneous equations Show our working. = + = + (f) Cop and complete this sentence: The line through D and B meets the line through G and H at (...,... ). (KS/9/Ma/Tier -/P). Equations in Contet In this section we determine the solutions to a variet of problems b forming and solving suitable linear equations. Eample Apples cost p per kg. Alan bus a bag of apples that costs.. If the bag contains kg of apples, (a) write down an equation involving, solve the equation.

37 (a) It is easier to work in pence. = = = = Eample Three consecutive whole numbers add up to. Determine the three numbers. If = first number, then and + = second number, + = third number. Adding these gives: + ( + ) + ( + ) = + = = = = and the three numbers are, and. Eample A tai driver charges.00 plus.0 per mile for all journes. (a) Write down the cost, in pence, for travelling m miles. The charge for a journe is.. Write down an equation and use this to determine the distance travelled. (a) Basic cost + 0 number of miles = m pence m = 0 m = 00 0 m =

38 . MEP Y9 Practice Book A m = 0 m =. So the distance travelled is. miles. Eercises. The cost of a ticket for a football match is 9. (a) Write down an epression for the cost of n tickets. Solve an equation to determine how man tickets could be bought with 0.. The cost of hiring a van is 0 per da, plus 0p for each mile travelled. (a) (c) Write down an epression for the cost, c, in pounds, of travelling m miles in one da in a hired van. Write down an epression for the cost in pounds of travelling m miles during a two-da hire period. James hires a van for das. He has to pa a total of.0. Write down an equation and solve it to determine how far he travelled.. Two consecutive odd numbers are and +. When these numbers are added together the total 00. Write down and solve an equation to obtain the value of.. A removals firm charges per mile plus a fied charge of. Use an equation to determine the distance travelled if the bill is 9.. The price of petrol is given in pence per litre. To convert this to per gallon, use the flow chart given below. Price in pence per litre. 00 Price in per gallon (a) (c) Convert a price of 0p per litre to per gallon. If the price is pence per litre, write down the cost in per gallon. Convert a price of. per gallon to pence per litre.. A rectangle has length 0 m and width m. (a) Write down a formula for the area of the rectangle. Use an equation to determine if the area is m. (c) Write down a formula for the perimeter of the rectangle. (d) Use an equation to determine, if the perimeter is 9 m.

39 . A repairman charges 0 for the first hour of his time and for each hour after that. (a) Write down a formula for the cost of a repair that takes n hours. Use an equation to determine the time for a repair, if the cost is.0.. At a bank a charge of is made for changing British Pounds ( ) into French Francs (Fr). The charge is deducted first and then 9 Fr are issued for ever left. (a) Write down a formula for the number of Fr issued in echange for. Use an equation to determine how man ou would need to change to get 900 Fr. 9. (a) Write down a formula for the perimeter of the shape shown. Calculate if the perimeter is. m. (c) (d) Write down a formula for the area of the shape. Calculate if the area is. m. 0. m (a) Write down a formula for the perimeter of the shape shown. If the perimeter is m, determine the length. m 9

40 . MEP Y9 Practice Book A. The simplified graph shows the flight details of an aeroplane travelling from London to Madrid, via Brussels. Madrid Distance from London (km) Brussels London (a) (c) Time (hours) GMT What is the aeroplane's average speed from London to Brussels? How can ou tell from the graph, without calculating, that the aeroplane's average speed from Brussels to Madrid is greater than its average speed from London to Brussels? A different aeroplane flies from Madrid to London, via Brussels. The flight details are shown below. Madrid depart 00 Brussels arrive 000 depart London arrive (d) On a cop of the graph, show the aeroplane's journe from Madrid to London, via Brussels. (Do not change the labels on the graph.) Assume constant speed for each part of the journe. At what time are the two aeroplanes the same distance from London? (KS/99/Ma/Tier -/P) 0

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