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 (xandy) , 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 1 : If it's easier, you can start by solving an equation for x in terms of y , also – same difference!

Example:

Solve the system { 3x+2y=16 7x+y=19

    Solve the second equation for y .

    y=197x

    Substitute 197x for y in the first equation and solve for x .

    3x+2(197x)=16 3x+3814x=16 11x=22 x=2

    Substitute 2 for x in y=197x and solve for y .

    y=197(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 .