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Chapter 2: Problem 59

Solve each equation, and check the solution. $$ -\frac{7}{9} x=\frac{3}{5} $$

Short Answer

Expert verified
The solution is \( x = -\frac{27}{35} \).

Step by step solution

01

Isolate the variable

To solve for the variable, multiply both sides of the equation by the reciprocal of the coefficient of the variable. The coefficient of the variable \(x\) is \(-\frac{7}{9}\), so multiply both sides by \(-\frac{9}{7}\). \(-\frac{7}{9} x \times -\frac{9}{7} = \frac{3}{5} \times -\frac{9}{7}\)
02

Solve the equation

Simplify both sides: \[ x = \frac{3}{5} \times -\frac{9}{7} \] Calculating the right-hand side: \[ x = \frac{3 \times -9}{5 \times 7} = \frac{-27}{35} \] Hence, \( x = -\frac{27}{35} \).
03

Check the solution

Substitute \( x = -\frac{27}{35} \) back into the original equation to verify: \[ -\frac{7}{9} \left(-\frac{27}{35}\right) = \frac{3}{5} \] Simplify the left-hand side: \[ \frac{7 \times 27}{9 \times 35} = \frac{189}{315} = \frac{3}{5} \] This confirms that \( x = -\frac{27}{35} \) is indeed the correct solution.

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Key Concepts

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

Reciprocal
To solve equations involving fractions, understanding the concept of reciprocals is crucial. A reciprocal of a number is simply 1 divided by that number. For example, the reciprocal of \(\frac{7}{9}\) is \(\frac{9}{7}\), and the reciprocal of -5 is \(-\frac{1}{5}\). When we multiply a number by its reciprocal, the result is always 1. For example: \(\frac{7}{9} \times \frac{9}{7} = 1\). This property is very useful when solving equations, as it allows us to cancel coefficients and isolate variables more easily.
Variable Isolation
Multiplying fractions is a straightforward process, but it鈥檚 essential to handle it correctly to solve equations accurately. When multiplying fractions, multiply the numerators (the numbers on the top) with each other and the denominators (the numbers on the bottom) with each other.

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