Homework Help: Solving Systems Of Equations By Substitution

How to Solve Systems of Equations

A 'system' typically includes two equations that have two variables in common (usually x and y). Finding a 'solution' to the system means you've found values for each of these variables that make both equations simultaneously true. These two values represent a coordinate point (x, y) where the lines representing the two equations intersect. Systems of equations have a finite number of solutions, infinite solutions or no solution at all.

The Substitution Method

The first step in using the substitution method is to solve one of the equations for one of the variables. It doesn't matter which equation you start with or which variable you solve for. This step allows you to state the value of one variable in terms of the other variable. Let's call the first equation 'Equation A' and the second equation 'Equation B.' For this illustration, we'll assume that the two variables in these equations are x and y, and that we've solved for x first.

The second step is to substitute the value of x in Equation A for x in Equation B. This transforms Equation B into a 1-variable equation, which allows you to solve for the numeric value of that variable (y). After you've done this, plug the value of that variable (y) back into Equation A and solve it for x.

Example

Here's a step-by-step example of how you can solve a system of equations using the substitution method. The two equations in the system are 3x - y = 1 and 3x + 2y = 16. First, we'll solve 3x - y = 1 for y:

3x - (3x) - y = - (3x) + 1

-y = -3x + 1

-y/-1 = (-3x + 1)/-1

y = 3x - 1

Next, we'll substitute 3x - 1 for the value of y in the other equation (3x + 2y = 16), and solve this equation for x.

3x + 2(3x - 1) = 16

3x + 6x - 2 = 16

9x - 2 = 16

9x - 2 + 2 = 16 + 2

9x = 18

9x/9 = 18/9

x = 2

Now that we know x = 2, we can plug two back into the first equation (3x - y = 1) and solve for y.

3(2) - y = 1

6 - y = 1

6 - 6 - y = 1 - 6

-y = -5

-y/-1 = -5/-1

y = 5

This tells you that the system of equations has the solution x = 2, y = 5 or (2, 5). Graphically, this means that the lines representing the two equations intersect at that point.