Solve linear equations #1 (2 variables - Algebra 1)

In the next 2 articles, I will attempt to explain how to solve a system of 2 linear equations.  The reason that you are asked to solve a system of 2 linear equations is to find out if the 2 lines will intersect or not.  The 2 linear lines can either intersect at one point, or the lines are parallel to each other, which means that there is no solution to the system of linear equations.

Algebra 1 students typically will be introduced to 2 different methods of solving the 2 linear equations.  The first method is known as substitution method, while the second method is known as the linear combination method.  This article we will explore the substitution method.

Since 2 linear equations means you have 2 equations, and 2 unknown variables.  As the word "substitution" implies, the whole method behind solving linear equations via substitution method is to solve one equation for one specific variable and substitute this variable in the second equation.  Then at this point, you will have 1 equation and one variable.

We will use the following linear equations for demonstration:

equation 1:  3x - 5y = 3
equation 2:  9x - 20y = 6

You can use equation 1 or 2 to solve for any variable first.  In this example, I will use equation 1 to solve for the variable x. 

equation 1:

3x - 5y = 3
3x = 3 + 5y
x = 1 + 5/3y  (becomes equation 1')

Substitute equation 1' back into equation 2.

equation 2:

9x - 20y = 6
9(1 + 5/3y) - 20y = 6
9 + 15y - 20y = 6
9 - 5y = 6
-5y = -3
y = 3/5

Substitute y = 3/5 into any one of the original equations such as equation 2.

9x - 20y = 6
9x - 20(3/5) = 6
9x  - 12 = 6
9x = 18
x = 2

Now that we have solved both variables, the answer for this linear equations is (2, 3/5).  In terms of graphing the 2 linear equations, it means that the 2 lines will intersect at the coordinates (2, 3/5).

Michael Huang
Center Director
Mathnasium of Glen Rock/Ridgewood
T:     201-444-8020   
E:  glenrock@mathnasium.com
www.mathnasium.com/glenrock