Solving Systems of Linear Equations by Multiplication and Addition

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.

The grid on the left still shows the same system, but the line $-x+3y=1$ has been temporarily highlighted in purple. 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. (Pick a value for $x$ and solve for $y$.)

Notice where the two points you just typed in have been plotted on the grid to the left. Are they on the purple line?

What can you conclude about the relationship between the graph of $-x+3y=1$ and the graph of your new equation? Are they the same or are they different?

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 (in standard form, $Ax+By=C$)?

Is the graph of this new equation the same as or different from the graph of $-x+3y=1$?

Use the slider to change the value of $a$. If you change $a$ to any number other than zero in the equation $a(-x+3y)=a(1)$, does it affect the graph?

If you change $a$ to 0 in the equation $a(-x+3y)=a(1)$, does it affect the graph?

Does multiplying a linear equation by a nonzero constant change its solution set?

You have seen the effects of multiplying an equation by a constant. Now let’s see what happens when you add together two equations, such as the ones to the right.

By adding two equations, we mean forming an equation whose left-hand side is the sum of the two equations’ left-hand sides, and whose right-hand side is the sum of the two equations’ right-hand sides.

The sum of the left-hand sides of the two equations above is the (unsimplified) expression $(-x+3y) + (6x+2y)$. What is the simplified form of this sum?

What is the sum of the two right-hand sides of the equations?

What equation is the result of adding $-x+3y=1$ and $6x+2y=9$?

Let’s look at the graph of this new equation together with the two original ones. 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 final equation you found in addEqs ($5x+5y=10$).

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?

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.

What does the system of equations become if you multiply the first equation by 2 (and rewrite it in standard form)?

What equation do you get when you add the two equations in this new system?

Use the sliders to change the values of $A$, $B$, and $C$ so that the blue line is the graph of the equation you just found. Do all three lines intersect at the same point?

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

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 mult2Add. 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.

→

Click to put the equation you just found into the system to the right. Then, add that equation to $6x+2y=9$.

What kind of line is the graph of the equation you just found? Is it horizontal, vertical, or neither?

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

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.

What is the point of intersection of the lines?

What is the solution to the system  ?

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.

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

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 that answer to $-5x+13y=-6$?

What is the solution to this system of equations?

As you have seen in the last few questions, you can solve systems of equations by eliminating a variable: that is, finding a way to add the equations together which makes that variable go away. In order to eliminate a variable by adding two equations together, you first have to multiply those equations by something so that the coefficients of that variable (the numbers multiplying that variable) are opposites.

Consider the system  , which is shown on the grid to the left. In this case, it is easier to multiply the second equation by some number $b$, in order to eliminate the $x$ terms.

What value of $b$ should we choose?

Click . Now the system is shown on the grid to the left. Sometimes it’s easier to multiply both equations by different numbers in order to eliminate the $x$ terms. We’ll multiply the first equation in the system by some number $a$, and the second equation by some other number $b$, to make the coefficients of $x$ be opposite.

If we let $b=2$ (that is, multiply the second equation in the system by 2), what value should we choose for $a$?

Now, we’ll solve the system from the last question, shown to the right.

What do you get if you multiply the first equation in the system by $-3$?

What do you get if you multiply the second equation in the system by 2?

Click to enter the equations you just found to the right. Notice that the two coefficients of $x$ are opposites. What is the result of adding those two equations?

? $+$ ?

What is the solution to this system of equations?