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!

    Example:

    Solve the system

    Solve the second equation for y.

                               y = 19 – 7x

    Substitute 19 – 7x for y in the first equation and solve for x.

                        3x + 2(19 – 7x) = 16

                          3x + 38 – 14x = 16

                                        –11x = –22

                                              x = 2

    Substitute 2 for x in y = 19 – 7x and solve for y.

                       y = 19 – 7(2)

                       y = 5

    The solution is (2, 5).

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.