Solving Systems of Linear Equations by Subtraction

The solution of a system of linear equations is a list of value(s) for the variable(s) that make all of the equations in the system true. Since every point on the graph of an equation solves that equation, the solution of a system of linear equations will be the intersection of the graphs of the equations. One of the methods for solving linear equations is using differences. In this lab you will learn about the difference of two equations and how to use this difference to find their point of intersection.

Question 1. Look at the diagram below. The point $(2,1)$ is on the line $y=x-1$. The point $(2,5)$ is on the line $y=5$.

To find the difference between the two lines when $x=2$, we can find the difference between the $y$-coordinates of these points. So, the difference is:

$5-1=4$.

Graph

What is the difference between the lines $y=5$ and $y=x-1$ when $x$ is 1?

Which line is above the other when $x$ is 1?

Which line is above the other when $x$ is 7?

Question 2. When $x$ is 8, the lines $y=5$ and $y=x-1$ are 2 units apart. To show that now $y=x-1$ is above $y=5$, we use the negative sign. So when $x$ is 8, the difference between $y=5$ and $y=x-1$ is −2.

Complete the table below. To complete the “difference” column, find the difference between $y=5$ and $y=x-1$ on the graph shown above.

$x$	$y=5$	$y=x-1$	difference $d$
1	5	0	5
2	5	1	4
3
5
6
7
8

What happens to the difference between the lines as you get closer and closer to the point of intersection?

What is the difference between the two lines at the point of intersection?

The red points on the grid to the left have coordinates $(x,d)$ from your table above. What do you notice about all of these red points?

We can find the equation for the line that goes through the red points by finding the difference between the two original equations. So, the equation for the line is

$d=5-(x-1)=5-x+1=6-x$.

Question 3. Click on to look at the graph of $d=6-x$ together with the lines $y=5$ and $y=x-1$. The red line should go through all of the red points.

What is the point of intersection of the original two lines (the blue lines)?

What is the $x$-intercept of $d$?

What is the relationship between the $x$-intercept of $d$ and the point of intersection of the original two lines?

Question 4. Each graph below is the graph of the difference between two lines. Find the $x$-coordinate of the point of intersection of the two lines. (Hint: See your answer to the question right before this one.)

$x=$

Question 5. Let's find the point of intersection of the lines $y=2x+1$ and $y=x-2$. First we will find the equation for $d$. To find $d$, we need to find the difference between the two equations:

$$\cl"ma-join-align"{\table d,=2x+1-(x-2); ,=2x+1-x+2; ,=x+3}$$

What is the $x$-coordinate of the $x$-intercept of $d=x+3$? (You can find the $x$-intercept of the equation by plugging in 0 for $d$.)

$x=$

What is the $x$-coordinate of the point of intersection of $y=2x+1$ and $y=x-2$? (Remember, this is where their difference, $d$, is 0.)

$x=$

Plug the $x$-coordinate of the point of intersection into $y=x-2$ to find the $y$-coordinate of the point of intersection.

$y=$

What is the point of intersection of the lines $y=2x+1$ and $y=x-2$?

Question 6. Click on to see the graphs of the equations $y=2x+1$, $y=x-2$ and $d=x+3$ from Question 5. Move the mouse to any place on the grid and click, hold, and drag the vertical bar to the left or right. To the right of the grid you will see the $y$ and $d$ values for each $x$ value you choose. For example when you move the bar to $x=4$, you will see $y=9,2$ and $d=7$. This means that when $x$ is 4, $y=9$ in the first equation, $y=2$ in the second equation, and, $d$, the difference between the lines, is 7.

Use the graph to find the point of intersection of the two lines $y=2x+1$ and $y=x-2$ (the blue lines).

Does your answer match your answer to the last part of Question 5?

Question 7. Complete the table below. Check your answers to make sure that the point of intersection you found satisfies both equations.
To fill in the second column, subtract the two equations as shown in question 5.
To get the values in the third column, plug $d=0$ into the equation in the second column.
To get the values in the fourth column, plug the value from the third column into either of the two original equations.

equations	difference $d$	$x$-value of point of intersection	$y$-value of point of intersection	Point of intersection
$y=2x-3$ $y=x+1$	$d=x-4$	4	5	$(4,5)$
$y=x+2$ $y=3x-2$
$y=5x$ $y=x+2$
$y=2x+1$ $y=2x-3$

In the last example above, is there a point of intersection? (Is $d$ ever 0?)

In that example, are the two lines parallel?

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