The substitution method for solving systems of equations is a way to simplify the system of equations by expressing one variable in terms of another, thus removing one variable from an equation. When the resulting simplified equation has only one variable to work with, the equation becomes solvable. 

The substitution method consists of the following steps:

1. In the first equation, solve for one of the variables in terms of the others.
2. Substitute this expression into the remaining equations.
3. Continue until you have reduced the system to a single linear equation.
4. Solve this equation, and then back-substitute until the solution is found.

Solving with the Substitution Method

Let's practice this by solving the following system of equations:

$x-y=-1$

$x+2y=-4$

We begin by solving the first equation so we can express x in terms of y. 

$\begin{aligned} \displaystyle x-y&=-1 \\x&=y-1 \end{aligned}$

Next, we will substitute our new definition of x into the second equation:

$\displaystyle \begin{aligned} x+2y&=-4 \\(y-1)+2y&=-4 \end{aligned}$

Note that now this equation only has one variable (y). We can then simplify this equation and solve for y:

$\displaystyle \begin{aligned} (y-1)+2y&=-4 \\3y-1&=-4 \\3y&=-3 \\y&=-1 \end{aligned}$

Now that we know the value of y, we can use it to find the value of the other variable, x. To do this, substitute the value of y into the first equation and solve for x.

$\displaystyle \begin{aligned} x-y&=-1 \\x-(-1)&=-1 \\x+1&=-1 \\x&=-1-1 \\x&=-2 \end{aligned}$

Thus, the solution to the system is: $(-2, -1)$, which is the point where the two functions graphically intersect. Check the solution by substituting the values into one of the equations.

$\displaystyle \begin{aligned} x-y&=-1 \\(-2)-(-1)&=-1 \\-2+1&=-1 \\-1&=-1 \end{aligned} $