SOLVING FOR A VARIABLE

COMBINING TERMS

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: -40x + 17 = 2x + 2075

Step 1 | If there are like terms on the same side of the equation, simplify the equation by combining the like terms. | There are no like terms on the same side of the equation to combine.
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. | -40x + 17 = 2x + 2075
| Eliminate the 2 x by adding -2x. | –40x +17 = 2x + 2075 -2x 00
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. | –40x + 17 = 2x + 2075 -2x 00
| Add -2x to 2x | -40x + 17 = 2x + 2075 -2x -2x -38x 0x
| Simplify the equation. | -40x + 17 = 2x + 2075 - 2x -2x -42x 0x -42x + 17 = 2075
Step 4 | Eliminate the non-coefficient numbers from the side of the equation containing the variable by performing inverse operations (opposite operations). | -42x + 17 = 2075
| The operation is addition. | -42x + 17 = 2075
| The inverse of addition is subtraction. Subtract 17 from 17. | -42x + 17 = 2075 -17 0
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. | -42x + 17= 2075 -17 0
| Subtract – 17 from 2075. | -42x + 17 = 2075 - 17 - 17 0 2058
| Simplify the equation. | -42x + 17 = 2075 -17 -17 0 2058 -42x = 2058
Step 6 | Eliminate the coefficient of the variable by performing the inverse operation. | -42x = 2058
| The operation is multiplication. | -42x = 2058
| The inverse of multiplication is division. Divide -42x by -42. | -42x = 2058 -42
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. | -42x = 2058 -42
| Divide 2058 by -42. | -42x = 2058 -42 -42
| Simplify the equation. | -42x = 2058 -42 -42 x = -49
Step 8 | Substitute the solution into the original equation to check the answer. | -40x + 17 = 2x + 2075 -40(-49) + 17 = 2(-49) + 2075 1960 + 17 = -98 + 2075 1977 = 1977