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Chapter 5: Problem 78

Solve and check. $$-\frac{3}{5} m=12$$

Short Answer

Expert verified
The solution to the equation is \(m = -20\)

Step by step solution

01

Isolate the variable

To isolate the variable 'm' on one side of the equation, divide both sides of the equation -\(\frac{3}{5} m = 12\) by -\(\frac{3}{5}\). The equation becomes \(m = 12/-\frac{3}{5}\). Remember that in division, dividing by a fraction is the same as multiplying by its reciprocal.
02

Simplify the right side of the equation

To simplify \(12/-\frac{3}{5}\), we multiply 12 by the reciprocal of -\(\frac{3}{5}\), which is -\(\frac{5}{3}\). This simplifies to \(m = -20\).
03

Check the solution

Substitute the value of 'm' back into the original equation. Thus, \(-\frac{3}{5}(-20) = 12\). Simplifying the left-hand side gives 12, which matches the right-hand side and confirms that the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolation of Variables
Isolating variables in equations is a fundamental skill in algebra that enables you to solve for the unknown quantity. The ultimate goal is to have the variable stand alone on one side of the equation, free from other numbers or coefficients. In this exercise, our equation is \(-\frac{3}{5} m = 12\).
To achieve isolation, we need to remove the fraction that is multiplied by \(m\). We do this by performing the opposite operation. Instead of multiplying, we'll divide both sides of the equation by \(-\frac{3}{5}\).
  • This division eliminates the fraction from the left side, leaving the variable \(m\) by itself.
  • The equation transforms to \(m = \frac{12}{-\frac{3}{5}}\).
Remember that when dividing by a fraction, you can multiply by its reciprocal, simplifying the process and leading us to the next crucial step.
Reciprocal Multiplication
Once the variable is isolated with a division, simplifying the resulting expression involves reciprocal multiplication. A reciprocal is what you multiply a number by to get 1. In the case of this exercise, to divide by \(-\frac{3}{5}\), you multiply by its reciprocal which is \(-\frac{5}{3}\).
Thus, the expression becomes:
  • Multiply \(12\) by \(-\frac{5}{3}\)
  • Perform the arithmetic: \(12 \times -\frac{5}{3} = -20\)
This process clarifies the new value of the isolated variable \(m\), which is \(-20\). Reciprocal multiplication is a handy technique that simplifies handling fractions when solving equations. It transforms division into an easier multiplication task.
Solution Verification
To ensure the solution you found is accurate, it is essential to verify it by substituting back into the original equation. Checking verifies not just the calculation but also boosts confidence in the result.

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