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Chapter 1: Problem 59

Solve. $$ \frac{1}{4} y-\frac{2}{3} y=5 $$

Short Answer

Expert verified
The solution is \( y = -12 \).

Step by step solution

01

Combine Like Terms

First, combine the terms involving \( y \) on the left side of the equation. To do this, you need a common denominator. The common denominator of 4 and 3 is 12. Convert each fraction: \begin{align*} \frac{1}{4} y & = \frac{3}{12} y \ \frac{2}{3} y & = \frac{8}{12} y. \text{Now combine these: } \frac{3}{12} y - \frac{8}{12} y = -\frac{5}{12} y \text{Hence, the equation becomes } -\frac{5}{12} y = 5. \text{}\end{align*}
02

Isolate the Variable

Next, solve for \( y \) by isolating the variable on one side of the equation. Multiply both sides by the reciprocal of the coefficient of \( y \), which is -\frac{12}{5}\begin{align*} -\frac{5}{12} y \cdot -\frac{12}{5}& = 5 \cdot -\frac{12}{5} \implies y = -12. \text{So, solving for } y \text{ yields } y = -12.\end{align*}

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Key Concepts

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

Combining Like Terms
To solve linear equations with fractions, it's important to first combine like terms. Like terms are terms that contain the same variable raised to the same power. For example, in the equation given \(\frac{1}{4} y - \frac{2}{3} y = 5\), the terms \(\frac{1}{4} y\) and \(\frac{2}{3} y\) are like terms because they both contain the variable \(y\). To combine them, find a common denominator. Converting each term to have this common denominator, you can then subtract one fraction from the other.
Common Denominators
When working with fractions, you often need to find a common denominator. The common denominator is a number that both denominators can divide into evenly. In our example, the denominators are 4 and 3. The smallest number that both 4 and 3 divide evenly into is 12.
Convert \(\frac{1}{4} y\) to \(\frac{3}{12} y\) and \(\frac{2}{3} y\) to \(\frac{8}{12} y\).
Combining these like terms gives \(\frac{3}{12} y - \frac{8}{12} y = -\frac{5}{12} y\).
Isolating the Variable
After combining like terms, the equation \(-\frac{5}{12} y = 5\) needs to be solved for \(y\). This involves isolating the variable \(y\). To do this, multiply both sides of the equation by the reciprocal of the coefficient of \(y\), which is \$ -\frac{12}{5} \$.
Performing this step: \(\begin{align*} \ -\frac{5}{12} y \ \cdot\ -\frac{12}{5} = 5 \ \cdot\ -\frac{12}{5} \ \implies\ y = -12\ \end{align*}\). Hence, \(y\) is isolated and found to be \( -12 \).

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Most popular questions from this chapter

Insert one pair of parentheses to convert each of the following false statements into a true statement. $$ 2 \cdot 7+3^{2} \cdot 5=104 $$

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Insert one pair of parentheses to convert each of the following false statements into a true statement. $$ 2-7 \cdot 2^{2}+9=-11 $$
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