Question

Solve the following equations for \(x.\) a. \(\frac{2}{3} x+\frac{1}{5} x=\frac{8}{15}\) b. \(\frac{1}{16} x-\frac{3}{4} x=\frac{1}{2}\) c. \(-\frac{1}{9} x-\frac{1}{3} x=\frac{1}{6}\) d. \(-\frac{1}{12} x-\frac{3}{4} x=-\frac{5}{36}\)

Short answer

a. \(x = \frac{8}{13}\), b. \(x = -\frac{8}{11}\), c. \(x = -\frac{3}{8}\), d. \(x = \frac{1}{6}\).

Step-by-step solution

Combine like terms for Equation a

Let's simplify the equation \( \frac{2}{3}x + \frac{1}{5}x = \frac{8}{15} \). First, find a common denominator for the fractions on the left-hand side, which is 15. Rewriting each term gives us: \( \frac{10}{15}x + \frac{3}{15}x = \frac{8}{15} \). Then, combine the terms to get \( \frac{13}{15}x = \frac{8}{15} \).

Solve for x in Equation a

To isolate \( x \), divide both sides by \( \frac{13}{15} \). This results in \( x = \frac{8}{15} \div \frac{13}{15} = \frac{8}{13} \). Thus, the solution for equation (a) is \( x = \frac{8}{13} \).

Combine like terms for Equation b

Consider the equation \( \frac{1}{16}x - \frac{3}{4}x = \frac{1}{2} \). Find a common denominator for \( \frac{1}{16} \) and \( \frac{3}{4} \), which is 16. Therefore, the equation becomes \( \frac{1}{16}x - \frac{12}{16}x = \frac{1}{2} \), simplifying to \( -\frac{11}{16}x = \frac{1}{2} \).

Solve for x in Equation b

To isolate \( x \), divide both sides by \( -\frac{11}{16} \). Thus, \( x = \frac{1}{2} \div -\frac{11}{16} = -\frac{8}{11} \). The solution for equation (b) is \( x = -\frac{8}{11} \).

Combine like terms for Equation c

Look at the equation \( -\frac{1}{9}x - \frac{1}{3}x = \frac{1}{6} \). The common denominator for \( \frac{1}{9} \) and \( \frac{1}{3} \) is 9. Rewrite the equation: \( -\frac{1}{9}x - \frac{3}{9}x = \frac{1}{6} \), which simplifies to \( -\frac{4}{9}x = \frac{1}{6} \).

Solve for x in Equation c

Solve by dividing both sides by \( -\frac{4}{9} \). So, \( x = \frac{1}{6} \div -\frac{4}{9} = -\frac{3}{8} \). The solution for equation (c) is \( x = -\frac{3}{8} \).

Combine like terms for Equation d

In the equation \( -\frac{1}{12}x - \frac{3}{4}x = -\frac{5}{36} \), the common denominator for \( \frac{1}{12} \) and \( \frac{3}{4} \) is 12. Rewriting gives \( -\frac{1}{12}x - \frac{9}{12}x = -\frac{5}{36} \), which simplifies to \( -\frac{10}{12}x = -\frac{5}{36} \), or \( -\frac{5}{6}x = -\frac{5}{36} \).

Solve for x in Equation d

To find \( x \), divide both sides by \( -\frac{5}{6} \). Therefore, \( x = -\frac{5}{36} \div -\frac{5}{6} = \frac{1}{6} \). The solution for equation (d) is \( x = \frac{1}{6} \).

Key concepts

Understanding Fractions

Fractions are an important part of dealing with prealgebra equations. They represent a part of a whole and consist of a numerator and a denominator. The numerator is the top number, indicating how many parts you have, while the denominator, as the bottom number, tells you how many parts the whole is divided into. In solving linear equations with fractions, understanding how to find a common denominator is crucial.

To add or subtract fractions, they must have the same denominator, known as the common denominator. For example, with the equation \( \frac{2}{3}x + \frac{1}{5}x = \frac{8}{15} \), the fractions on the left have different denominators. By converting each fraction to have a common denominator, here 15, we can proceed to combine the fractions into a single term.

• Find the least common multiple (LCM) of the denominators.
• Adjust each fraction to have the LCM as the new denominator.
• Add or subtract the numerators, keeping the denominator the same.

In our example, \( \frac{2}{3}x \) becomes \( \frac{10}{15}x \), and \( \frac{1}{5}x \) turns into \( \frac{3}{15}x \), allowing us to combine them into \( \frac{13}{15}x \).

Solving Linear Equations

Solving linear equations is a foundational skill in algebra. A linear equation is any equation that can be written in the form \( ax + b = c \), where \(a\), \(b\), and \(c\) are constants. The goal of solving the equation is to find the value of \(x\).

To solve these equations, you usually need to manipulate the equation in order to isolate \(x\). This involves a series of steps such as combining like terms or changing the fraction.

• Start by simplifying each side of the equation: apply operations like addition, subtraction, or factoring to make it less complex.
• Use operations like addition, subtraction, multiplication, or division to isolate the variable on one side.

In equation (a), after combining like terms, you have \( \frac{13}{15}x = \frac{8}{15} \). To isolate \(x\), divide both sides by \( \frac{13}{15} \) to find \( x = \frac{8}{13} \). This method can be applied to any of the equations provided.

Combining Like Terms

Combining like terms is an essential technique for simplifying and solving equations. Like terms are terms that contain the same variable raised to the same power. Only like terms can be combined.

For example, in \( \frac{2}{3}x + \frac{1}{5}x \), both terms involve the variable \(x\), making them like terms. The rule is to add or subtract the coefficients (numbers in front of \(x\)) while keeping the variable the same.

• Identify terms with the same variable.
• Rewrite those terms with a common denominator if necessary (especially important if fractions are involved).
• Add or subtract the coefficients of these terms.

Applying this to our example, once \( \frac{2}{3}x \) and \( \frac{1}{5}x \) are given a common denominator, they become \( \frac{10}{15}x \) and \( \frac{3}{15}x \). These can then be added to form \( \frac{13}{15}x \). This makes the equation simpler and prepares it for solving, underscoring the power of combining like terms effectively.