Analytic Geometry

Ontario 09 Academic (MPM1D)

Solving equations with straight lines

Lesson

In algebra, we've looked at how to solve different types of equations. Now we are going to look at how to find a solution that is true for two equations simultaneously, which, when we graph it, is call the point of intersection. A point of intersection is the point where two lines meet or cross. There are many different ways to solve equations, so let's get going and look at some of these different methods.

We can solve equations graphically by plotting two straight lines on a number plane and finding their point of intersection. For example, as you'll see in the Question 1 below, we have plotted two equations: $y=4x+1$`y`=4`x`+1 and $y=-x+5$`y`=−`x`+5. We graph the lines to find the coordinates of the point of intersection, which is $\left(1,4\right)$(1,4). This would be the solution to these equations.

What is the point of intersection? State your answer in coordinate form $\left(a,b\right)$(`a`,`b`).

Consider the lines:

$x=-1$`x`=−1

$y=-2x+2$`y`=−2`x`+2

a) Complete the table of values for $y=-2x+2$`y`=−2`x`+2.

b) Graph both lines.

c) What is their point of intersection? State your answer in coordinate form $\left(a,b\right)$(`a`,`b`).

We don't always need to graph the lines to find the point of intersection. We can also just use tables of values to find them. We just need to look for the corresponding $x$`x` and $y$`y` values in both tables.

For example, below are tables of values for $y=-3x+4$`y`=−3`x`+4 and $y=2x-1$`y`=2`x`−1. Notice how both tables have $\left(1,1\right)$(1,1) in them? This means this is our point of intersection and the solution to our equations!

Just be careful though! The point of intersection may have really big numbers, in which case we would need giant tables of values! If you don't notice one pretty quickly, you're probably better off graphing the lines.

Find the point of intersection for $y=-3x+4$`y`=−3`x`+4 and $y=2x-1$`y`=2`x`−1.

$y=-3x+4$`y`=−3`x`+4

$x$x |
$0$0 | 1 | $2$2 | $3$3 |

$y$y |
$4$4 | 1 | $-2$−2 | $-5$−5 |

$y=2x-1$`y`=2`x`−1

$x$x |
$-1$−1 | $0$0 | 1 | $2$2 |

$y$y |
$-3$−3 | $-1$−1 | 1 | $3$3 |

The point of intersection is $\left(1,1\right)$(1,1).

We can also solve equations and find these points of intersection algebraically but we'll look more at that later.

Determine graphically the point of intersection of two linear relations, and interpret the intersection point in the context of an application