How to Solve Math Linear Equations

Substitution and Elimination

• 1

  Use the substitution or elimination methods to solve a linear equation system. For example, use substitution in the following system:

  3x+y=4 (first equation)

  2x-2y=7 (second equation)

• 2

  Modify the first equation to be y = 4-3x. Substitute "4-3x" into place of "y" in the second equation to get 2x-2(4-3x)=7. Distribute the 2 over the parenthesis to get 2x-8+6x=7. Simplify terms to get 8x -8 = 7 and 8x = 15. Therefore x = 15/8 = 1.875.

• 3

  Plug in the x-value into either original equation. The first equation becomes 3(1.875) + y = 4. Combine like terms as before to get y = 4 -- 3(1.875) and y = -1.625.

• 4

  Use elimination to solve the example equations. Multiply each term of the first equation by 2: 2(3x) + 2(y) = 2(4) is 6x + 2y = 8. Note that there is a positive "2y" in the first equation and a "-2y" in the second (unmodified) equation.

• 5

  Add matching terms of both equations. For 6x+2y=8 and 2x-2y=7, the addition is 6x+2x, 2y + (-2y), 8 + 7. The result is 8x + 0y = 15, or just 8x = 15. As before, x = 15/8 = 1.875, so y = -1.625. Obtaining identical answers using different methods is a way to check that the solution (x and y values) to the equation system is correct.

Matrix Manipulation

• 6

  Rewrite the linear equations in matrix form. Matrix format allows easier variable manipulation. In addition, matrixes are not as "cluttered" or intimidating.

• 7

  Manipulate matrix rows to produce row echelon and then a reduced row-echelon matrix. Add, subtract, multiply, divide and rearrange rows (but not columns) to get reduced row-echelon form. In the example, 3x+y=4 becomes row 1: R1 = [3 1 4]. Similarly, 2x-2y=7 is row 2: R2 = [2 -2 7]. Row-echelon form gives rows [1 0 1.875] and [0 1 -1.625]. The first row indicates that the 1x + 0y = x = 1.875 and 0x + 1y = y = -1.625. Once again, the result matches previous solutions.

• 8

  Program an algorithm to convert a linear equation into matrixes. Use coding loops and conditional statements ("if" statements) to convert a matrix from original to row-echelon form. In systems with dozens or hundreds of equations and variables, matrix-based linear algebra is a time saver.