SOLVING FOR A VARIABLE

RATIOS

Concept: An equation is a mathematical statement that shows that two quantities are equal. When an equation contains a variable, we use algebra to find the value of the variable. To solve an equation means to find the number which replaces the variable to make the mathematical statement true.

The basic strategy in solving an equation with a variable is to isolate the variable, that is, get the variable alone on one side of the equation and the constant terms (numbers) on the other side of the equation. We do this by combining like terms and performing inverse (opposite) operations. Addition and subtraction are inverse operations. Multiplication and division are inverse operations.

Steps To Use When Solving For A Variable

Step 1 | Simplify the equation by combining like terms. If there are like terms on the same side of the equation, simplify the equation by combining the like terms.
Step 2 | If there are like variables on both sides of the equation, eliminate the variable on one side of the equation using addition/subtraction operations.
Step 3 | Whatever is done on one side of the equation in Step 2 must also be done on the other side of the equation now.
Step 4 | Eliminate the non-coefficient numbers from the side of the equation containing the variable by performing inverse operations (opposite operations).
Step 5 | Whatever is done on one side of the equation in Step 4 must also be done on the other side of the equation now.
Step 6 | Eliminate the coefficient of the variable by performing the inverse operation.
Step 7 | Whatever is done on one side of the equation in Step 6 must also be done on the other side of the equation now.
Step 8 | Substitute the solution into the original equation to check the answer.

Example Solve for x:

Step 1 | Simplify the equation by combining like terms. If there are like terms on the same side of the equation, simplify the equation by combining the like terms. | Does not apply to this equation.
Step 2 | If there are like variables on both sides of the equation, eliminate the variable on one side of the equation using addition/subtraction operations. | Does not apply to this equation.
Step 3 | Whatever is done on one side of the equation in Step 2 must also be done on the other side of the equation now. | Does not apply to this equation.
Step 4 | Eliminate the non-coefficient numbers from the side of the equation containing the variable by performing inverse operations (opposite operations). | Does not apply to this equation.
Step 5 | Whatever is done on one side of the equation in Step 4 must also be done on the other side of the equation now. | Does not apply to this equation.
Step 6 | Eliminate the coefficient of the variable by performing the inverse operation. |
| The operation is division. | 15 x - 5 13
| The inverse of division is multiplication. Multiply by 13. | 13
Step 7 | Whatever is done on one side of the equation in Step 6 must also be done on the other side of the equation now. | 1 3
| Multiply by 13. | 13 13
| Simplify the equation. | 1 3 13 39 = x
Step 8 | Substitute the solution into the original equation to check the answer. Check your answer by cross-multiplying. | 195 = 195

Another way to do the same Example Solve for x:

Step 1 | Cross-multiply. | 195 = 5x
Step 2 | Eliminate the coefficient of the variable by performing the inverse operation. | 195 = 5x
| The operation is multiplication. | 195 = 5x
| The inverse of multiplication is division. Divide 5x by 5. |
=
195 5x 5
Step 3 | Whatever is done on one side of the equation in Step 2 must also be done on the other side of the equation now. |
=
195 5x 5
| Divide 195 by 5. |
=
195 5x 5 5
| Simplify the equation. |
=
195 5x 5 5 39 = x