Solving Systems of Linear Equations Using Substitution

Systems of Linear equations:

A system of linear equations is just a set of two or more linear equations.

In two variables (x and y), the graph of a system of two equations is a pair of lines in the plane.

There are three possibilities:

• The lines intersect at zero points. (The lines are parallel.)
• The lines intersect at exactly one point. (Most cases.)
• The lines intersect at infinitely many points. (The two equations represent the same line.)

How to Solve a System Using The Substitution Method

• Step 1: First, solve one linear equation for y in terms of x.
• Step 2: Then substitute that expression for y in the other linear equation. You'll get an equation in x.
• Step 3: Solve this, and you have the x-coordinate of the intersection.
• Step 4: Then plug in x to either equation to find the corresponding y-coordinate.

Note: If it's easier, you can start by solving an equation for x in terms of y, also – same difference!

Note 2: If the lines are parallel, your x-terms will cancel in step 2, and you will get an impossible equation, something like 0 = 3.

Note 3: If the two equations represent the same line, everything will cancel in step 2, and you will get a redundant equation, 0 = 0.