10.06 The point of intersection
Solving by graphing
A linear graph will extend forever in both directions. This means that two unique lines must intersect at some point, unless the two lines have the same gradient and are therefore parallel. We can use this fact to solve equations graphically by plotting two straight lines on a number plane and finding the coordinates of the point where the two lines cross. This is known as the point of intersection.
We can use this method to solve simple equations like $4x=8$4x=8 and $3+x=8$3+x=8, and also much harder equations, containing variables on both sides such as $2x-7=-8x+13$2x−7=−8x+13. Let's explore how we can solve these $3$3 equations graphically.
Worked examples
EXAMPLE 1
Solve the equation $4x=8$4x=8 by finding the point of intersection of the two lines $y=4x$y=4x and $y=8$y=8.
Think: To sketch a straight line we only need to find two points on it. Looking at $y=4x$y=4x first, if we substitute in $x=0$x=0 and $x=1$x=1, we will get $y=0$y=0 and $y=4$y=4 respectively so we know the line will pass through the two points $\left(0,0\right)$(0,0) and $\left(1,4\right)$(1,4). The line $y=8$y=8 is a horizontal line that passes through the point $0,8$0,8.
Do: We can now sketch both lines.
First, let's sketch the line for $y=4x$y=4x.
We can now sketch the line $y=8$y=8 on the same number plane as follows;
The point of intersection of the two lines is shown, and has the coordinates $\left(2,8\right)$(2,8).
This means the solution to $4x=8$4x=8 is $x=2$x=2, as the $x$x-coordinate of the point of intersection is $2$2.
Reflect: By sketching both lines on the same number plane we can easily see the point where the two lines intersect. Since the point of intersection gives us the same $y$y-value for both the equations $y=4x$y=4x and $y=8$y=8, we know that the $x$x-coordinate of the point of intersection makes $4x$4x and $8$8 equal.
We can verify our solution algebraically: if $4x=8$4x=8 dividing both sides by $2$2 easily gives the solution $x=4$x=4.
While this was a relatively simple example, we can use this method to solve more complicated equations, such as the one below.
EXAMPLE 2
Solve the equation $4x-2=-3x+12$4x−2=−3x+12 by first finding the point of intersection of the two lines $y=4x-2$y=4x−2 and $y=-3x+12$y=−3x+12.
Think: Looking at $y=4x-2$y=4x−2 first, if we substitute in $x=0$x=0 and $x=1$x=1, we will get $y=-2$y=−2 and $y=2$y=2 respectively so we know the line will pass through the two points $\left(0,-2\right)$(0,−2) and $\left(1,2\right)$(1,2). Substituting $x=0$x=0 and $x=1$x=1 into $y=-3x+12$y=−3x+12 gives us the points $\left(0,12\right)$(0,12) and $\left(1,9\right)$(1,9).
Do: We can now sketch both lines on the same number plane.
The point of intersection of the two lines is shown, and has the coordinates $\left(2,6\right)$(2,6). This means the solution to the equation $4x-2=-3x+12$4x−2=−3x+12 is $x=2$x=2.
Reflect: We can verify this solution algebraically:
| $4x-2$4x−2 | $=$= | $-3x+12$−3x+12 |
|
| $4x-2+3x$4x−2+3x | $=$= | $12$12 |
Add $3x$3x to both sides of the equation |
| $7x-2$7x−2 | $=$= | $12$12 |
Combine the $x$x-terms |
| $7x$7x | $=$= | $14$14 |
Add $2$2 to both sides of the equation |
| $x$x | $=$= | $2$2 |
Divide both sides of the equation by $7$7 |
Solving algebraically gives us the same solution $x=2$x=2.
Practice Questions
question 1
What is the point of intersection? State your answer in coordinate form $\left(x,y\right)$(x,y).
Question 2
In this question we will find the point of intersection between the line $y=2x+2$y=2x+2 and the line $x=-3$x=−3.
Fill in the blanks to complete the table of values for $y=2x+2$y=2x+2:
$x$x $-1$−1 $0$0 $1$1 $2$2 $y$y $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Now sketch the line $y=2x+2$y=2x+2, together with the line $x=-3$x=−3:
Loading Graph...What is the point of intersection of these two lines?
State your answer in coordinate form $\left(x,y\right)$(x,y).
Question 2
In this question we will find the point of intersection between the line $y=x-9$y=x−9 and the line $y=-x-7$y=−x−7.
Sketch the line $y=x-9$y=x−9, together with the line $y=-x-7$y=−x−7:
Loading Graph...What is the point of intersection of these two lines?
State your answer in coordinate form $\left(x,y\right)$(x,y).