What does it mean to solve a system of equations? Think about what it means to solve a single algebraic equation. For instance, to solve the equation $3x+1=7$, we need to find the value of $x$ such that the left-hand side of the equation equals
when we plug $x$ in. With a system of equations, we want to find the values of all of the variables such that all of the equations are true.
Since we know how to solve a linear equation with only one variable, one method for solving a system of linear equations is to eliminate one of the variables. Let's look at how to do this for our example.
We'll try to eliminate $x$ first. One method to do so is to first multiply each equation by a constant so that the coefficients of $x$ in the two equations are the same. There are many ways to do this, but the simplest choice is to multiply each equation by the coefficient in the other equation. If we do this, the first equation should be multiplied by
and the second equation should be multiplied by
. The first equation is now
and the second equation is now
Now that the coefficients of $x$ match, we can eliminate $x$ by subtracting one equation from the other. We'll subtract the second equation from the first. What is the resulting equation?
With $x$ eliminated, we can solve for $y$. We find that $y=$
.
The next step is to use $y$ to find $x$. We can use any of the equations to do this. We'll use the first equation (from above or from the beginning of the problem). We plug in the value of $y$ we found in part a to get an equation just involving $x$:
We can now solve this for $x$, which tells us that $x=$
.
To summarize, the solution to the system of equations is $x=$
, $y=$
, which we can write in vector form as $\vc{x}=$
.