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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 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 { 3 x + 2 y = 16 7 x + y = 19

    Solve the second equation for y .

    y = 19 7 x

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

    3 x + 2 ( 19 7 x ) = 16 3 x + 38 14 x = 16 11 x = 22 x = 2

    Substitute 2 for x in y = 19 7 x 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 .