1. 8: Simultaneous Equations and Intersections © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

2. Simultaneous Equations and Intersections Module C1 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

3. Simultaneous Equations and Intersections Suppose we want to find where 2 lines meet. The point of intersection has an x -value between -1 and 0 and a y -value between 3 and 4. e.g. 1 and Sketching the lines gives The exact values can be found by solving the equations simultaneously

4. Simultaneous Equations and Intersections As the y -values are the same, the right-hand sides of the equations must also be the same. At the point of intersection, we notice that the x -values on both lines are the same and the y - values are also the same. Substituting into one of the original equations, we can find y : The point of intersection is

5. Simultaneous Equations and Intersections Sometimes the equations first need to be rearranged: Substituting into (1): The point of intersection is Solution: Equation (2) can be written as Now, eliminating y between (1) and (2a) gives: e.g. 2

6. Simultaneous Equations and Intersections Exercises 2. 1. Point of intersection is Eliminate y : Rearrange (1): Eliminate y : Find the point of intersection of the following pairs of lines: Solution:

7. Simultaneous Equations and Intersections There are 2 points of intersection We again solve the equations simultaneously but this time there will be 2 pairs of x- and y- values e.g. 3 Find the points of intersection of and

8. Simultaneous Equations and Intersections Since the y -values are equal we can eliminate y by equating the right hand sides of the equations: This is a quadratic equation, so we get zero on one side and try to factorise: To find the y- values, we use the linear equation, which in this example is equation (2) The points of intersection are (1, 1) and (-3, 9) e.g. 1

9. Simultaneous Equations and Intersections e.g. 2 Sometimes we need to rearrange the linear equation before eliminating y Rearranging (2) gives Eliminating y : or Substituting in (2a):

10. Simultaneous Equations and Intersections Exercise Find the points of intersections of the following curve and line The solution is on the next slide

11. Simultaneous Equations and Intersections Eliminate y : Substitute in ( 2a ): Solution: The points of intersection are Rearrange (2):

12. Simultaneous Equations and Intersections Solving the equations simultaneously will not give any real solutions Special Cases e.g. 1 Consider the following equations: The line and the curve don’t meet.

13. Simultaneous Equations and Intersections The quadratic equation has no real roots. Suppose we try to solve the equations: Eliminate y : Calculating the discriminant, we get:

14. Simultaneous Equations and Intersections e.g. 2 Eliminate y : The discriminant, The quadratic equation has equal roots. The line is a tangent to the curve. Solving

15. Simultaneous Equations and Intersections  To solve a linear and a quadratic equation simultaneously: SUMMARY Solve for the 2 nd unknown Substitute into the linear equation to find the values of the 1 st unknown. 2 points of intersection the line is a tangent to the curve the line and curve do not meet and the equations have no real solutions.  A linear and a quadratic equation represent a line and a curve. Eliminate one unknown to give a quadratic equation in the 2 nd unknown, e.g.

16. Simultaneous Equations and Intersections Exercises Decide whether the following pairs of lines and curves meet. If they do, find the point(s) of intersection. For each pair, sketch the curve and line. 1. 2. 3.

17. Simultaneous Equations and Intersections Solutions 1. the line is a tangent to the curve

18. Simultaneous Equations and Intersections Solutions 2.there are 2 points of intersection

19. Simultaneous Equations and Intersections Solutions 3.there are NO points of intersection

20. Simultaneous Equations and Intersections

21. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

22. Simultaneous Equations and Intersections As the y -values are the same, the right- hand sides of the equations must also be the same. Two Lines At the point of intersection, we notice that the x -values on both lines are the same and the y -values are the same. Substituting into one of the original equations, we can find y : The point of intersection is

23. Simultaneous Equations and Intersections 1 quadratic equation and 1 linear equation e.g. This is a quadratic equation, so we get zero on one side and try to factorise: To find the y- values, we use the linear equation, which in this example is equation (2) The points of intersection are (1, 1) and (-3, 9) Since the y -values are equal we can eliminate y by equating the right hand sides of the equations:

24. Simultaneous Equations and Intersections e.g. Sometimes we need to rearrange the linear equation before eliminating y Rearranging (2) gives Eliminating y : or Substituting in (2a):

25. Simultaneous Equations and Intersections Solving the equations simultaneously will not give any real solutions. Special Cases e.g. 1 Consider the following equations: The line and the curve don’t meet. The discriminant

26. Simultaneous Equations and Intersections e.g. 2 Eliminate y : The discriminant, The quadratic equation has equal roots. The line is a tangent to the curve. Solving

27. Simultaneous Equations and Intersections  To solve a linear and a quadratic equation simultaneously: SUMMARY Solve for the 2 nd unknown Substitute into the linear equation to find the values of the 1 st unknown. 2 points of intersection the line is a tangent to the curve the line and curve do not meet and the equations have no real solutions.  A linear and a quadratic equation represent a line and a curve. Eliminate one unknown to give a quadratic equation in the 2 nd unknown, e.g.