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In this lesson you will learn how to find solutions to systems of linear equations algebraically, and you’ll see graphically why the method works.
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?
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 $-2x+6y=2$.
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.
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 ($5x+5y=10$).
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.
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.
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.
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.