CHAPTER 35 GRAPHICAL SOLUTION OF EQUATIONSdocuments.routledge-interactive.s3.amazonaws.com/...y xx= −−412 and y = 0 coincide give the solution of 4 10xx2 −−= . This is seen

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CHAPTER 35 GRAPHICAL SOLUTION OF EQUATIONS

EXERCISE 143 Page 369

1. Solve the simultaneous equations graphically: y = 3x – 2

y = – x + 6 Since both equations represent straight-line graphs, only two coordinates plus one to check are

needed. x 0 1 2 x 0 1 2

y = 3x – 2 – 2 1 4 y = –x + 6 6 5 4

Both graphs are shown plotted below.

The solution of the simultaneous equations is where the two graphs intersect

From the graphs the solution is x = 2, y = 4 2. Solve the simultaneous equations graphically: x + y = 2

3y – 2x = 1 x + y = 2 from which, y = –x + 2

3y – 2x = 1 from which, 3y = 2x + 1 and y = 2 13 3

x +

Both graphs are shown plotted below.

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The solution of the simultaneous equations is where the two graphs intersect From the graphs the solution is x = 1, y = 1 3. Solve the simultaneous equations graphically: y = 5 – x

x – y = 2 Since both equations represent straight-line graphs, only two coordinates plus one to check are

needed. x 0 1 2 x 0 1 2

y = –x + 5 5 4 3 y = x – 2 – 2 – 1 0

Both graphs are shown plotted below.

The solution of the simultaneous equations is where the two graphs intersect

From the graphs the solution is x = 3.5, y = 1.5

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4. Solve the simultaneous equations graphically: 3x + 4y = 5

2x – 5y + 12 = 0

3x + 4y = 5 from which, 4y = –3x + 5 and y = 3 54 4

x− +

2x – 5y + 12 = 0 from which, 5y = 2x + 12 and y = 2 125 5

x +

Both graphs are shown plotted below. The solution of the simultaneous equations is where the two graphs intersect From the graphs the solution is x = –1, y = 2

5. Solve the simultaneous equations graphically: 1.4x – 7.06 = 3.2y

2.1x – 6.7y = 12.87

1.4x – 7.06 = 3.2y hence, y = 1.4 7.063.2 3.2

x − i.e. y = 0.4375x – 2.20625

x 0 1 2

y – 2.21 – 1.77 – 1.33

2.1x – 6.7y = 12.87 hence, y = 2.1 12.876.7 6.7

x − i.e. y = 0.31343x – 1.9209

x 0 1 2

y – 1.92 – 1.61 – 1.29 Both graphs are shown plotted below The solution of the simultaneous equations is where the two graphs intersect

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From the graphs the solution is x = 2.3, y = –1.2

6. Solve the simultaneous equations graphically: 3x – 2y = 0

4x + y + 11 = 0

3x – 2y = 0 from which, y = 32

x

4x + y + 11 = 0 from which, y = – 4x – 11 Both graphs are shown plotted below.

The solution of the simultaneous equations is where the two graphs intersect From the graphs the solution is x = –2, y = –3

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7. The friction force F newtons and load L newtons are connected by a law of the form F = aL + b, where a and b are constants. When F = 4 N, L = 6 N and when F = 2.4 N, L = 2 N. Determine graphically the values of a and b. Since F = aL + b, then 4 = 6a + b from which, b = –6a + 4 and 2.4 = 2a + b from which, b = –2a + 2.4 Both graphs are shown plotted below

The solution of the simultaneous equations is where the two graphs intersect From the graphs the solution is a = 0.4, b = 1.6

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EXERCISE 144 Page 373

1. Sketch the following graphs and state the nature and coordinates of their turning points:

(a) y = 4x2 (b) y = 2x2 – 1 (c) y = –x2 + 3 (d) y = – 12

x2 – 1

(a) A graph of y = 4x2 is shown below. The turning point is a minimum at (0, 0)

(b) A graph of y = 2x2 – 1 is shown below. The turning point is a minimum at (0, –1)

(c) A graph of y = –x2 + 3 is shown below. The turning point is a maximum at (0, 3)

(d) A graph of y = – 12

x2 – 1 is shown below. The turning point is a maximum at (0, –1)

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2. Solve graphically the quadratic equation 4x2 – x – 1 = 0 by plotting the curve between the limits

x = –1 to x = 1. Give answers correct to 1 decimal place.

A graph of 24 1y x x= − − is shown below

The points where 24 1y x x= − − and y = 0 coincide give the solution of 24 1 0x x− − = . This is seen to be at x = –0.4 and x = 0.6 3. Solve graphically the quadratic equation x2 – 3x = 27 by plotting the curve between the limits

x = –5 to x = 8. Give answers correct to 1 decimal place.

A graph of 2 3 27y x x= − − is shown below The points where 2 3 27y x x= − − and y = 0 coincide give the solution of 2 3 27x x− = . This is seen to be at x = –3.9 and x = 6.9

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4. Solve graphically the quadratic equation 2x2 – 6x – 9 = 0 by plotting the curve between the

limits x = –2 to x = 5. Give answers correct to 1 decimal place.

A graph of 22 6 9y x x= − − is shown below

The points where 22 6 9y x x= − − and y = 0 coincide give the solution of 22 6 9 0x x− − = . This is seen to be at x = –1.1 and x = 4.1 5. Solve graphically the quadratic equation 2x(5x – 2) = 39.6 by plotting the curves between the

limits x = –2 to x = 3. Give answers correct to 1 decimal place.

2x(5x – 2) = 39.6 i.e. 210 4 39.6 0x x− − = A graph of y = 210 4 39.6x x− − is shown below

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The points where y = 210 4 39.6x x− − and y = 0 coincide give the solution of 210 4 39.6x x− − = 0 This is seen to be at x = –1.8 and x = 2.2 6. Solve the quadratic equation 2x2 + 7x + 6 = 0 graphically, given that the solutions lie in the range x = –3 to x = 1. Determine also the nature and coordinates of its turning point. A graph of 22 7 6y x x= + + is shown below The points where 22 7 6y x x= + + and y = 0 coincide give the solution of 22 7 6 0x x+ + = . This is seen to be at x = –1.5 and x = –2 The turning point is a minimum and occurs at x = –1.75 and the minimum value is y = –0.1

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7. Solve graphically the quadratic equation 10x2 – 9x – 11.2 = 0, given that the roots lie between x = –1 and x = 2 A graph of y = 10x2 – 9x – 11.2 is shown below

The points where y = 10x2 – 9x – 11.2 and y = 0 coincide give the solution of 10x2 – 9x – 11.2 = 0 This is seen to be at x = –0.7 and x = 1.6 8. Plot a graph of y = 3x2 and hence solve the equations: (a) 3x2 – 8 = 0 (b) 3x2 – 2x – 1 = 0 A graph of 23y x= is shown below

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(a) 23 8 0x − = hence, 23 8x = . The solution of 23 8 0x − = is determined from the intersection of 23y x= and y = 8 and from the above graphs this occurs at x = –1.63 and x = 1.63 (b) 23 2 1 0x x− − = hence, 23 2 1x x= + . The solution of 23 2 1 0x x− − = is determined from the intersection of 23y x= and y = 2x + 1 and from the above graphs this occurs at x = –0.3 and x = 1 9. Plot the graphs y = 2x2 and y = 3 – 4x on the same axes and find the coordinates of the points of intersection. Hence determine the roots of the equation 2x2 + 4x – 3 = 0 A graph of 22y x= is shown below Also, a graph of y = 3 – 4x is shown on the same axes. The coordinates of the points of intersection of the two graphs occurs at (–2.6, 13.2) and (0.6, 0.8)

22 4 3 0x x+ − = is equivalent to 22 3 4x x= − The solution of 22 4 3 0x x+ − = is determined from the intersection of 22y x= and y = 3 – 4x and from the above graphs, the roots of 22 4 3 0x x+ − = are x = –2.6 and x = 0.6

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10. Plot a graph of y = 10x2 – 13x – 30 for values of x between x = – and x = 3. Solve the equation 10x2 – 13x – 30 = 0 and from the graph determine: (a) the value of y when x is 1.3, (b) the value of x when y is 10 and (c) the roots of the equation 10x2 – 15x – 18 = 0 A graph of 210 13 30y x x= − − is shown below. When y = 0, the solution of 210 13 30 0x x− − = is x = –1.2 and x = 2.5 (a) From the graph, when x = 1.3, y = –30 (b) From the graph, when y = 10, x = –1.50 and x = 2.75 (c) 210 15 18 0x x− − = is equivalent to 210 13 30 2 12x x x− − = − Thus the roots of the equation 210 15 18 0x x− − = occurs at the intersection of the graphs 210 13 30y x x= − − and y = 2x – 12, i.e. the roots are x = 2.3 and x = –0.8

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EXERCISE 145 Page 374

1. Determine graphically the values of x and y which simultaneously satisfy the equations: y = 2(x2 – 2x – 4) and y + 4 = 3x

( )2 22 2 4 2 4 8y x x x x= − − = − − y + 4 = 3x i.e. y = 3x – 4 The two graphs are shown plotted below.

The points of intersection provide the solutions to the simultaneous equations, i.e. x = 4, y = 8 and x = –0.5, y = –5.5 2. Plot the graph of y = 4x2 – 8x – 21 for values of x from –2 to +4. Use the graph to find the roots of the following equations: (a) 4x2 – 8x – 21 = 0 (b) 4x2 – 8x – 16 = 0 (c) 4x2 – 6x – 18 = 0 A graph of 24 8 21y x x= − − is shown plotted below

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(a) From the graph, the roots of 24 8 21 0x x− − = are x = –1.5 and x = 3.5

(b) 24 8 16 0x x− − = is equivalent to 24 8 21 5x x− − = −

Hence, the intersection of the graphs 24 8 21y x x= − − and y = –5 gives the roots of the

equation 24 8 16 0x x− − = , i.e. x = –1.24 and x = 3.24

(c) 24 6 18 0x x− − = is equivalent to 24 8 21 2 3x x x− − = − −

Hence, the intersection of the graphs 24 8 21y x x= − − and y = –2x – 3 gives the roots of the

equation 24 6 18 0x x− − = , i.e. x = –1.5 and x = 3.0

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EXERCISE 146 Page 375

1. Plot the graph y = 4x3 + 4x2 – 11x – 6 between x = –3 and x = 2 and use the graph to solve the cubic equation 4x3 + 4x2 – 11x – 6 = 0 A graph of 3 24 4 11 6y x x x= + − − is shown plotted below.

The solution of 3 24 4 11 6 0x x x+ − − = is determined from the intersection of

3 24 4 11 6y x x x= + − − and y = 0 and from the above graph, the roots of 3 24 4 11 6 0x x x+ − − = are x = –2.0, x = –0.5 and x = 1.5 2. By plotting a graph of y = x3 – 2x2 – 5x + 6 between x = –3 and x = 4 solve the equation x3 – 2x2 – 5x + 6 = 0. Determine also the coordinates of the turning points and distinguish between them.

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A graph of 3 22 5 6y x x x= − − + is shown plotted below.

The solution of 3 22 5 6 0x x x− − + = is determined from the intersection of

3 22 5 6y x x x= − − + and y = 0 and from the above graph, the roots of 3 22 5 6 0x x x− − + = are x = –2, x = 1 and x = 3 The turning points are: minimum at (2.1, –4.1) and maximum at (–0.8, 8.2) 3. Solve graphically the cubic equation: x3 – 1 = 0 correct to 2 significant figures. Let 3 1y x= − A table of values is shown below x –2 –1 0 1 2 3

y –9 –2 –1 0 7 26 A graph of 3 1y x= − is shown below and it can be seen that there is only one root for the equation

3 1 0x − = , i.e. the root is at x = 1

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4. Solve graphically the cubic equation: x3 – x2 – 5x + 2 = 0 correct to 2 significant figures. A graph of y = 3 2 5 2x x x− − + is shown below

The solution of 3 2 5 2 0x x x− − + = is determined from the intersection of y = 3 2 5 2x x x− − + and y = 0 and from the above graph, the roots of 3 2 5 2 0x x x− − + = are x = –2.0, x = 0.4 and x = 2.6 5. Solve graphically the cubic equation: x3 – 2x2 = 2x – 2 correct to 2 significant figures x3 – 2x2 = 2x – 2 hence, x3 – 2x2 – 2x + 2 = 0 A graph of y = x3 – 2x2 – 2x + 2 is shown below

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The solution of x3 – 2x2 – 2x + 2 = 0 is: x = –1.2, 0.70 and 2.5 6. Solve graphically the cubic equation: 2x3 – x2 – 9.08x + 8.28 = 0 correct to 2 significant figures. A graph of y = 2x3 – x2 – 9.08x + 8.28 is shown below

The solution of 2x3 – x2 – 9.08x + 8.28 = 0 is: x = –2.3, 1.0 and 1.8 7. Show that the cubic equation 8x3 + 36x2 + 54x + 27 = 0 has only one real root and determine its value.

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Let 3 28 36 54 27y x x x= + + + A table of values is shown below x –3 –2 –1 0 1 2

y –27 –1 1 27 125 343 A graph of 3 28 36 54 27y x x x= + + +

The solution of 3 28 36 54 27 0x x x+ + + = is determined from the intersection of 3 28 36 54 27y x x x= + + + and y = 0 and from the above graph, the one and only root of

3 28 36 54 27 0x x x+ + + = is x = –1.5 (It can also be seen from the above table of values that there is only one root to the equation)