Problem 26 - Precalculus Student Solutions Manual 5th

Question

Solve each equation. $$\frac{2}{3} x+0.25 x=x+2$$

Step-by-Step Solution

Answer

x = -24

Step 1 - Combine like terms on the left side

Combine \(\frac{2}{3}x\) and \(0.25x\) on the left side of the equation. First, convert \(0.25x\) to a fraction: \(0.25 = \frac{1}{4} \), so the equation becomes \(\frac{2}{3} x + \frac{1}{4} x = x + 2\).

Step 2 - Find a common denominator

To combine \(\frac{2}{3}x\) and \( \frac{1}{4}x \), find a common denominator. The least common multiple of 3 and 4 is 12. Convert \(\frac{2}{3}x\) and \( \frac{1}{4}x \) to have the same denominator: \(\frac{2}{3}x = \frac{8}{12}x\) and \( \frac{1}{4}x = \frac{3}{12}x \).

Step 3 - Add the fractions

Now that the fractions have the same denominator, add \(\frac{8}{12} x \) and \(\frac{3}{12}x \): \(\frac{8}{12} x + \frac{3}{12} x = \frac{11}{12} x \). The equation is now \(\frac{11}{12} x = x + 2 \).

Step 4 - Isolate the variable x

Subtract \(x\) from both sides to isolate \(x\): \(\frac{11}{12} x - x = 2 \).

Step 5 - Simplify both sides

Recognize that \(\frac{11}{12} x - x\) is \(\frac{11}{12} x - \frac{12}{12} x = -\frac{1}{12} x \). Thus, the equation becomes \(-\frac{1}{12} x = 2 \).

Step 6 - Solve for x

Multiply both sides by \(-12\) to solve for \(x\). This gives: \(-\frac{1}{12} x \times -12 = 2 \times -12 \), simplifying to \ x = -24 \.

Key Concepts

solving linear equations

Linear equations look like this: \(ax + b = c\). They involve variables like \(x\) and constants like \(b\) and \(c\). The goal is to find the value of the variable that makes the equation true. We start by performing operations like adding, subtracting, multiplying, or dividing to isolate the variable. In our example, we have \(\frac{2}{3} x + 0.25 x = x + 2\). First, we simplify the equation by combining like terms.

combining like terms

Like terms have the same variable raised to the same power. In \(\frac{2}{3}x + 0.25x\), both terms include \(x\) raised to the power of 1. To combine them, first convert the decimal to a fraction: \(0.25x = \frac{1}{4} x\). Now the equation is \(\frac{2}{3}x + \frac{1}{4}x = x + 2\). Combining these results in adding the coefficients: \(\frac{2}{3}\) and \(\frac{1}{4}\).

finding common denominators

When adding fractions, we need a common denominator. For \(\frac{2}{3}x\) and \(\frac{1}{4}x\), the least common denominator (LCD) is 12. Convert each term: \(\frac{2}{3}x = \frac{8}{12}x\) and \(\frac{1}{4}x = \frac{3}{12}x\). Now, both fractions have the same denominator and can be added: \(\frac{8}{12} x + \frac{3}{12} x = \frac{11}{12} x\). The equation becomes \(\frac{11}{12} x = x + 2\).

fraction addition

Adding fractions means summing the numerators if the denominators are common. In our case: \(\frac{8}{12} x + \frac{3}{12} x = \frac{11}{12} x\). This simplifies the equation to \(\frac{11}{12} x = x + 2\). Next, we need to isolate the variable \(x\) by moving terms involving \(x\) on one side of the equation.

isolating variables

To isolate \(x\), subtract \(x\) from both sides: \(\frac{11}{12} x - x = 2\). Simplify: \(\frac{11}{12} x - \frac{12}{12} x = -\frac{1}{12} x = 2\). Now, solve for \(x\) by dividing both sides by \(-\frac{1}{12}\) or multiply by \(-12\): \(-\frac{1}{12} x \times -12 = 2 \times -12\). Solving gives \(x = -24\). This completes the solution.