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

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

Substituting into one of the original equations, we can find y: 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. As the y-values are the same, the right-hand sides of the equations must also be the same. Substituting into one of the original equations, we can find y: The point of intersection is

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

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

e.g. 3 Find the points of intersection of and 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

The points of intersection are (1, 1) and (-3, 9) e.g. 1 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)

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

Exercise Find the points of intersections of the following curve and line The solution is on the next slide

Solution: Rearrange (2): Eliminate y: Substitute in (2a): The points of intersection are

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

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

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

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

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.

Solutions 1. the line is a tangent to the curve

Solutions 2. there are 2 points of intersection

Solutions 3. there are NO points of intersection

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.

Substituting into one of the original equations, we can find y: 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

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:

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

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

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

2 points of intersection To solve a linear and a quadratic equation simultaneously: SUMMARY Solve for the 2nd unknown Substitute into the linear equation to find the values of the 1st 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 2nd unknown, e.g.