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The coefficient of is To solve for we need its coefficient to be Since the product of a number and its reciprocal is our strategy here will be to isolate by multiplying by the reciprocal of We will do this in [link] .
Solve:
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| Multiply both sides by the reciprocal of the coefficient. |
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| Simplify. |
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| Multiply. |
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| Check: |
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| Substitute . |
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| Rewrite as a fraction. |
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| Multiply. The equation is true. |
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Notice that in the equation we could have divided both sides by to get by itself. Dividing is the same as multiplying by the reciprocal, so we would get the same result. But most people agree that multiplying by the reciprocal is easier.
Solve:
The coefficient is a negative fraction. Remember that a number and its reciprocal have the same sign, so the reciprocal of the coefficient must also be negative.
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| Multiply both sides by the reciprocal of . |
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| Simplify; reciprocals multiply to one. |
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| Multiply. |
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| Check: |
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| Let . |
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| Multiply. It checks. |
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Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.
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Subtraction Property of Equality:
For any real numbers and if then |
Addition Property of Equality:
For any real numbers and if then |
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Division Property of Equality:
For any numbers and where if then |
Multiplication Property of Equality:
For any real numbers and if then |
When you add, subtract, multiply or divide the same quantity from both sides of an equation, you still have equality.
In the next few examples, we’ll translate sentences into equations and then solve the equations. It might be helpful to review the translation table in Multiply Whole Numbers .
Translate and solve: divided by is
| Translate. |
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| Multiply both sides by . |
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| Simplify. |
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| Check: | Is divided by equal to ? | |
| Translate. |
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| Simplify. It checks. |
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Translate and solve: The quotient of and is
| Translate. |
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| Multiply both sides by . |
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| Simplify. |
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| Check: | Is the quotient of and equal to ? | |
| Translate. |
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| Simplify. It checks. |
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Translate and solve: Two-thirds of is
| Translate. |
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| Multiply both sides by . |
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| Simplify. |
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| Check: | Is two-thirds of equal to ? | |
| Translate. |
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| Simplify. It checks. |
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Translate and solve: The quotient of and is
| The quotient of and is . | |
| Translate. | |
| Multiply both sides by to isolate . | |
| Simplify. | |
| Remove common factors and multiply. | |
| Check: | |
| Is the quotient of and equal to ? | |
| Rewrite as division. | |
| Multiply the first fraction by the reciprocal of the second. | |
| Simplify. |
Our solution checks.
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