Solving Systems of Linear Equations by Graphing

A system of equations is a set 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 lab you will learn about solutions to systems of linear equations and how to find them by graphing.
$$\{\,\cl"ma-join1-align"{\table x+2y=-7; 2x-3y=0}$$

Solutions of equations

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

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

Use the method shown above to decide if each of the points in the table to the right is a solution of the equation $x+2y=4$.
PointIs the point a solution?
$(2,1)$Yes
$(1,1)$
$(0,2)$
$(4,0)$
$(-1,4)$

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.

Are the points that are on the line solutions?

Are the points that aren't on the line solutions?

Name one more point that is a solution.

Question 2. Do you think that all of the points on the line are solutions to $x+2y=4$?

In general, what can you say about points on a line and the solutions of an equation for the line?

Question 3. Click on to see the graphs of $x+3y=6$ and $x-y=2$.

Where do the two lines intersect?

Plug the coordinates of this point into the equation $x-y=2$. Is it a solution?

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

You have just solved this system of equations and you have found that the solution of this system is $(3,1)$.
$$\{\,\cl"ma-join1-align red"{\table x+3y=6; x-y=2}$$

Question 4. 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?

Question 5. Click on to see the graphs of the equations in the system shown here.
$$\{\,\cl"ma-join1-align red"{\table x+2y=-7; 2x-3y=0}$$

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 it a solution?

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

Question 6. Click on . 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.
$$\{\,\cl"ma-join1-align red"{\table Ax+By=C; x+y=-1}$$
 

Question 7. Find an $Ax+By=C$ equation so that the system of equations has no solutions.
$$\{\,\cl"ma-join1-align red"{\table Ax+By=C; x+y=-1}$$
 

Question 8. Find an $Ax+By=C$ equation so that the system of equations has infinitely many solutions.
$$\{\,\cl"ma-join1-align red"{\table Ax+By=C; x+y=-1}$$
 

Question 9. Click on to see 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.
$$\{\,\cl"ma-join1-align blue"{\table x+y=-1; 2x+y=1; -x+2y=-8}$$

What is the solution for the system?

Plug the coordinates of the point you found into the first equation, $x+y=-1$. Is it a solution?

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, $-x+2y=-8$?

Question 10. Click on to see the graphs of the equations in the system shown here.
$$\{\,\cl"ma-join1-align blue"{\table x+y=-1; 2x+y=1; x-y=0}$$

Does this system have a solution?

If there is a solution, name the point. If not, explain why not.

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