Solving Systems of Linear Equations by Graphing

A system of equations is a list of two or more equations using the same variables. For example, the pair of equations shown to the right is a system of linear equations. In this lesson you will learn about solutions to systems of linear equations and how to find them by graphing.
$$\{\,\cl"tight"{\table x,+,2y,=,-7; 2x,-,3y,=,0}$$

Solutions to equations

A solution to an equation is a list of value(s) for the variable(s) that make the equation true. For example, $(x,y)=(2,1)$ is a solution to the equation $x+2y=4$:

$x+2y$ $=$ $4$
$x=2$ and $y=1$, so $2+2(1)$ $=$ $4$
$2+2$ $=$ $4$
$4$ $=$ $4$

Use the method shown above to decide if each of the points in the table below is a solution to the equation $x+2y=4$.

Point $(x, y)$Is the point a solution?

Each of the points from the table above is plotted on the grid to the left. The graph of $x+2y=4$ is also shown on the grid.

Which two points from the table are solutions to the equation?
Are those points on the line which is the graph of $x+2y=4$?
Which two points from the table are not solutions to the equation?
Are those points on the line which is the graph of $x+2y=4$?
Name one more point that is a solution to the equation.
Do you think that all of the points on the line are solutions to $x+2y=4$?
Do you think that any of the points not on the line are solutions to $x+2y=4$?

Solutions to systems of two equations

To the left are the graphs of $x+3y=6$ and $x-y=2$.

At which point do the two lines intersect?

Plug the coordinates of this point into the equation $x-y=2$.

Is the point of intersection a solution to the equation $x+3y=6$?

You have just solved this system of equations: you have found that the solution to the system is the point $(3,1)$.
$$\{\,\cl"tight red"{\table x,+,3y,=,6; x,-,y,=,2}$$

A solution to a system of equations is a single list of values that is a solution to all of the equations.

Can you find any other points that are solutions to both $x+3y=6$ and $x-y=2$?
How many solutions does this system of equations have?
In how many places do the lines intersect?
To the left are the graphs of the equations in the system shown here.
$$\{\,\cl"tight red"{\table x,+,2y,=,-7; 2x,-,3y,=,0}$$
How many solutions does this system of equations have?
What is/are the solution(s) to this system?

Plug the coordinates of this point into the equation $x+2y=-7$.

Is the point you found a solution to the equation $2x-3y=0$?

Use the sliders to change the values of $A$, $B$, and $C$. Find an $Ax+By=C$ equation so that the system of equations shown here has one solution, and type that equation in the input box to the right.
$$\{\,\cl"tight red"{\table Ax,+,By,=,C; x,+,y,=,-1}$$
Find an $Ax+By=C$ equation so that the system of equations has no solutions.
$$\{\,\cl"tight red"{\table Ax,+,By,=,C; x,+,y,=,-1}$$
Find an $Ax+By=C$ equation so that the system of equations has infinitely many solutions.
$$\{\,\cl"tight red"{\table Ax,+,By,=,C; x,+,y,=,-1}$$

Solutions to systems of three equations

To the left are the graphs of the equations in the system shown here. In this exercise you will solve this system of equations. A solution to this system is a point that satisfies all three equations. This means it’s the point where the three lines intersect.
$$\{\,\cl"tight blue"{\table x,+,y,=,-1; 2x,+,y,=,1; -x,+,2y,=,-8}$$
What is the solution for the system?

Plug the coordinates of the point you found into the first equation, $x+y=-1$.

Is the point you found a solution for the second equation, $2x+y=1$?

Is the point you found a solution for the third equation, $-x+2y=-8$?

To the left are the graphs of the equations in the system shown here.
$$\{\,\cl"tight blue"{\table x,+,y,=,-1; 2x,+,y,=,1; x,-,y,=,0}$$
Is there any single point where all three of these graphs intersect?
Does this system have a solution?