Problem 55 - Algebra for College Students

Question

Solve each equation, and check the solution. \(-\frac{5}{9} x=2\)

Step-by-Step Solution

Answer

The solution is \( x = -\frac{18}{5} \).

Step 1 - Isolate the Variable

To isolate the variable, we need to get rid of the fraction. Multiply both sides of the equation by the reciprocal of \(\frac{-5}{9}\), which is \(\frac{-9}{5}\).

Step 2 - Perform the Multiplication

Multiply each side of the equation by \(\frac{-9}{5}\): \(\frac{-9}{5} \times -\frac{5}{9}x = \frac{-9}{5} \times 2\).

Step 3 - Simplify the Equation

Simplify both sides: \((\frac{-9 \times -5}{9 \times 5})x = -\frac{18}{5}\) which simplifies to \((1)x = -\frac{18}{5}\) or \(x = -\frac{18}{5}\).

Step 4 - Check the Solution

Substitute \(x = -\frac{18}{5}\) back into the original equation to verify: \(-\frac{5}{9} \times -\frac{18}{5} = 2\). Simplifying the left-hand side, we get \[ \frac{90}{45} = 2 \] which simplifies to \[ 2 = 2 \]. Therefore, the solution is correct.

Key Concepts

Isolate the Variable

The first step to solve any linear equation is to isolate the variable (usually represented by letters like x or y). This means rearranging the equation so that the variable stands alone on one side of the equation. In our example, the equation was \(-\frac{5}{9} x = 2\). To isolate x, we need to eliminate the fraction. Multiplying both sides of the equation by the reciprocal of \(-\frac{5}{9}\) helps achieve this.

Reciprocal

The reciprocal of a fraction is simply flipping it. For instance, the reciprocal of \(-\frac{5}{9}\) is \(-\frac{9}{5}\). When you multiply a fraction by its reciprocal, the result is 1. This property is used to clear fractions from equations. In our exercise, multiplying both sides of the equation by \(-\frac{9}{5}\) is crucial because it helps in making sure the variable (x) is left alone. \[\frac{-9}{5} \times -\frac{5}{9}x = \frac{-9}{5} \times 2\] simplifies to \[(1)x = -\frac{18}{5}\] or \x = -\frac{18}{5}\.

Simplifying Fractions

To solve linear equations, it is often necessary to simplify fractions. Once you multiply both sides by the reciprocal, you might end up with complex fractions. Simplify these fractions by canceling out common factors in the numerator and the denominator. For example, \[-\frac{9 \times -5}{9 \times 5}\] simplifies neatly to 1 because the numerator and denominator are the same. Knowing how to simplify fractions is a fundamental skill in solving algebraic equations.

Checking Solutions in Algebra

After solving an equation, it's important to check your solution to ensure it's correct. This involves substituting the solution back into the original equation and verifying that both sides are equal. In our exercise, we substitute \x = -\frac{18}{5}\ back into the original equation: \-\frac{5}{9} \times -\frac{18}{5} = 2\. Simplifying, we get \[\frac{90}{45} = 2\] which simplifies to \2 = 2\. Therefore, the solution \x = -\frac{18}{5}\ is correct. Checking solutions ensures there are no errors in calculations and verifies that the isolated variable is indeed the true solution.