Problem 26 - Algebra for College Students

Question

Find the solution set to each equation. $$\frac{5}{x}=\frac{7}{9}$$

Step-by-Step Solution

Answer

The solution set is \(x = \frac{45}{7}\).

Step 1: Cross-multiply

To solve the equation \(\frac{5}{x} = \frac{7}{9}\), start by cross-multiplying to eliminate the fractions. Multiply 5 by 9 and 7 by x: \[5 \times 9 = 7 \times x\]

Step 2: Simplify the equation

Simplify the expression obtained in Step 1. You get: \[45 = 7x\]

Step 3: Solve for x

Isolate x by dividing both sides of the equation by 7: \[x = \frac{45}{7}\]

Key Concepts

Cross-Multiplication

Cross-multiplication is a method used to solve rational equations, especially those involving fractions. When you have an equation in the form of \(\frac{a}{b} = \frac{c}{d}\), you can eliminate the fractions by cross-multiplying.
This means you multiply the numerator of one fraction by the denominator of the other fraction.
Following this method, for the equation \(\frac{5}{x} = \frac{7}{9}\), you multiply 5 (numerator of the first fraction) by 9 (denominator of the second fraction) and 7 (numerator of the second fraction) by x (denominator of the first fraction):
\[5 \times 9 = 7 \times x\]
This gives you an equation without fractions, making the next steps simpler.

Isolating Variables

Isolating variables is a technique used to solve equations by getting the variable by itself on one side of the equation. After cross-multiplying in the equation \(\frac{5}{x} = \frac{7}{9}\), you will have:
\[45 = 7x\]
To isolate x, you need to divide both sides of the equation by 7. This step is crucial to solve for x:
\[7x = 45\]
\[x = \frac{45}{7}\]
In this way, you have isolated x and found its value. Always perform the same operation on both sides of the equation to maintain equality.

Fractions in Algebra

Fractions in algebra can often make an equation look more complicated.
However, using proper techniques like cross-multiplication can simplify them. When dealing with fractions, it helps to understand concepts such as:

• Numerators and denominators - these are the top and bottom parts of a fraction, respectively.
• Cross-multiplication - useful for solving equations with fractions on both sides.
• Equivalent fractions - understanding that \(\frac{a}{b} = \frac{c}{d}\) implies \a \times d = b \times c\.

In the equation \(\frac{5}{x} = \frac{7}{9}\), recognizing that the fractions can be eliminated through cross-multiplication simplifies the problem significantly.
By converting the fractions to a statement without them, you can easily solve for the variable.