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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 lesson you will learn about the difference of two
equations and how to use this difference to find their point of intersection.
Look at the diagram below or the grid to the left. 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:
When $x$ is 8, the lines $y=5$ and
$y=x-1$ are 2 units apart. At the differences we computed
in the previous question, the line $y=5$
was above the line $y=x-1$. To show that now
$y=x-1$ is above $y=5$, we use a 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
the graphs of $y=5$ and $y=x-1$ on the grid
to the left.
You 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:
Look at the graph of $d=6-x$ together with the lines
$y=x-1$. Notice that the red line
goes through all of the red points.
On the grid to the left is the graph of the difference between two lines. Find
the $x$-coordinate of the point of intersection of the two lines and enter it into the table
below. Do this for each row of the table. (See your answer to the question
right before this one.)
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:
Here are the graphs of the equations
$d=x+3$ from solveQn.
Click or tap inside the grid and drag 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 $y=9$, $y=2$, and $d=7$.
This means that, when $x$ is 4:
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
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.
Up until now, we’ve been solving systems of equations in
slope-intercept form ($y=mx+b$). You can also use this method to solve
systems of equations in standard form ($Ax+By=C$) by first converting
those equations into slope-intercept form.
Convert the equation $4x+2y=6$ from standard form into slope-intercept form.