Solving Systems of Linear Equations by Multiplication & Addition

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 how to find solutions to systems of linear equations algebraically, and you'll see graphically why the method works.
$$\{\,\cl"ma-join1-align"{\table -x+3y=1; 6x+2y=9}$$

Question 1. On the grid to the left you are looking at the graphs of $-x+3y=1$ and $6x+2y=9$. Can you find the exact coordinates of the point of intersection of these lines just by looking at the graph?

In cases like this, when the exact location of the point of intersection can't be found using the graph alone, we have to use other methods to find the solution to a system of equations.

We will look at one of these methods which is sometimes referred to as the method of "Multiplication and Addition". Let's see how and why this method works.

Question 2. What equation do you get when you multiply both sides of the equation $-x+3y=1$ by 2?
$2(-x+3y)=2(1)$
→ 

Find two points that satisfy the equation you just found. (Hint: pick a value for $x$ and solve for $y$.)  

Notice that the two points you just selected are plotted on the grid to the left. What do you notice about the points? What can you conclude about the relationship between the graph of $-x+3y=1$ and the graph of your new equation?

Question 3. Click on . So far you have seen that multiplying an equation by 2 doesn't affect its graph. What about multiplying it by 3, or 5, or −1? Let's use the grid to the left to help answer this question. The grid shows the graph of $a(-x+3y)=a(1)$. The value of $a$ is set to 2, which means that each side of the equation $-x+3y=1$ is being multiplied by 2, so the graph you are looking at is the graph of $-2x+6y=2$.

Using the slider, set the value of $a$ to 3. What is the equation for the graph you're looking at now?

How does the graph of this new equation compare to the graph of $-x+3y=1$?

Use the slider to change the value of $a$. What do you notice? Does changing $a$ in the equation $a(-x+3y)=a(1)$ affect the graph?

In general, what can you say about the effect of multiplying a linear equation by a nonzero constant?

Question 4. You have seen the effects of multiplying an equation by a constant. Now let's see what happens when you add two equations together. What equation is the result of adding $-x+3y=1$ and $6x+2y=9$?
$-x+3y=1$
$+$$6x+2y=9$

Question 5. Let's look at the graph of this new equation together with the two original ones. Click on and use the sliders to change the values of $A$, $B$, and $C$ in the equation $Ax+By=C$ so that the blue line is the graph of the sum you found in Question 4 ($5x+5y=10$).

Do all three lines intersect at the same point?

Can you find the exact coordinates of the point of intersection just by looking at the graph?

Question 6. You have seen that multiplying an equation by a constant doesn't change its graph, so let's multiply the first equation in this system by 2.
$$\{\,\cl"ma-join1-align"{\table -x+3y=1; 6x+2y=9}$$

$2(-x+3y)=2(1)$ reduces to $-2x+6y=2$. So, the system of equations becomes:
$$\{\,\cl"ma-join1-align"{\table -2x+6y=2; 6x+2y=9}$$

Adding these two equations, we get:
$-2x+6y=2$
$+$$6x+2y=9$
$4x+8y=11$

Use the sliders to change the values of $A$, $B$, and $C$ so that the blue line is the graph of $4x+8y=11$. Do all three lines intersect at the same point?

Can you find the exact coordinates of this point just by looking at the graph?

Question 7. Click on . We have multiplied the first equation by $a$, and added the second equation, to form the third equation you see (the equation for the blue line). The value of the constant $a$ is now set to 2. This means the first equation is multiplied by 2 and added to the second equation, just like in Question 6. Use the slider to change the value of $a$ and notice the effect on the graph.

Does changing $a$ affect the point of intersection of the three lines?

What value of $a$ makes the blue line horizontal?

Multiply the first equation in this system by your answer to the previous question.
$[\,\,](-x+3y)=[\,\,](1)$
→ 

Add the equation you just found to $6x+2y=9$.
$-6x+18y=6$
$+$$6x+2y=9$

What happened? What kind of line is the graph of the equation you found in the previous question?

Find the $y$-coordinate of the point of intersection by solving the last equation you found. $y=$

Question 8. Now that you know the $y$-coordinate of the point of intersection of the lines, you can find the $x$-coordinate by plugging the $y$-coordinate into either of the original equations.

Plug the $y$-coordinate of the point of intersection into either of the original equations and find the $x$-coordinate. $x=$

What is the point of intersection of the three lines?

What is the solution to the system?
$$\{\,\cl"ma-join1-align"{\table -x+3y=1; 6x+2y=9}$$
$x=$  $y=$

Question 9. Click on . You are looking at the graph of the system of linear equations shown to the right. The equation for the blue line is again formed by multiplying the first equation by $a$, and adding the second equation.
$$\{\,\cl"ma-join1-align"{\table x+3y=4; -5x+13y=-6}$$

Use the slider to change the value of $a$. What $a$ value should you use to start solving this system?

What equation do you get when you multiply both sides of $x+3y=4$ by the value of $a$ you found?

What equation do you get when you add your answer to the previous question to $-5x+13y=-6$?

What is the solution to this system of equations? $x=$  $y=$

Does changing $a$ affect the point of intersection of the three lines?

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