How to Solve Linear Equations of College Algebra

One-Variable Linear Equations

• 1

  Recall inverse relationships, such as 1 and -1, and 1/3 and 3, to solve one-variable linear equations. The solutions require using the inverse relationships of addition and subtraction, and multiplication and division.

• 2

  Isolate the variable "x" on one side of the equation. If x = y, then x + a = y + a. Based on this logic, use inverses to move values from one side of an equation to the other side of the equation.

  To isolate x by using the inverse of subtraction in the equation x - 5 = 8, add the inverse of -5, which is +5, to both sides of the equation. The result is: x - 5 + 5 = 8 + 5. The solution is: x = 13.

  To use the inverse of addition in the equation x + 9 = 12 to isolate x, subtract the inverse of +9 from both sides of the equation. The resulting equation is: x + 9 - 9 = 12 - 9. After subtracting 9 from both sides of the equation, you will find that x = 3.

  Using the inverse of division in the equation (1/2)x = 10 to isolate x requires multiplying the inverse of 1/2 by both sides of the equation. The resulting equation is: (1/2)(2) = 10(2). Multiplying both sides of the equation by 2 reveals that x = 20.

  To isolate x by using the inverse of multiplication in the equation 4x = 8, divide both sides of the equation by 4. The resulting equation is: 4x/4 = 8/4. The solution is: x = 2.

• 3

  Check the solution. Plug the solution into the original equation to verify that its value is correct. If the original equation is x - 5 = 8 and you found that the value of x is 13, for instance, then check the solution by simply using the value 13 instead of x in the original equation. The equation then becomes 13 -- 5 = 8 or 8 = 8, which is the correct answer.

Two-Variable Linear Equations -- Addition/Elimination Method

• 4

  Choose a variable to eliminate in a two-variable linear equation such as 4x -- 10y = 32 and 6x + 4y = 10. To eliminate "x," multiply the equations by common multiples to obtain equal but opposite values of x: 3(4x -- 10y = 32) and -2(6x + 4y = 10). The example will then look like this: 12x -- 30y = 96 and -12x -- 8y = -20.

• 5

  Add the equations together to eliminate x. An example is:

  12x -- 30y = 96

  -12x -- 8y = -20

  _____________

  -38y = 76

• 6

  Solve for y in the equation -38y = 76. The process is:

  -38y/38 = 76/38

  -y = 2

  -y/-1 = 2/-1

  y = -2

• 7

  Plug the value of y into the original equations, and find the value for x. The first original equation is 4x -- 10y = 32, and the solution process is:

  4x -- 10(-2) = 32

  4x + 20 = 32

  4x + 20 -- 20 = 32 -- 20

  4x = 12

  4x/4 = 12/4

  x = 3

  The second original equation is 6x + 4y = 10. Its solution process is:

  6x + 4(-2) = 10

  6x -- 8 = 10

  6x -- 8 + 8 = 10 + 8

  6x = 18

  6x/6 = 18/6

  x = 3

• 8

  Check the solutions y = -2 and x = 3 for the original equations, 4x -- 10y = 32 and 6x + 4y = 10. The process for the first equation is:

  4(3) -- 10(-2) = 32

  12 +20 = 32

  32 = 32

  The process for the second equation is:

  6(3) + 4(-2) = 10

  18 -- 8 = 10

  10 = 10

  Two-variable linear equations can have one solution, no solution or many solutions. That is why it is very important to check solutions in the original equations.