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Chapter 2: Problem 330

Solve each equation with fraction coefficients. $$\frac{1}{3} w+\frac{5}{4}=w-\frac{1}{4}$$

Short Answer

Expert verified
The solution is \[ w = \frac{9}{4} \]

Step by step solution

01

Eliminate the fractions

To eliminate the fractions, find the least common multiple (LCM) of the denominators (3 and 4). The LCM of 3 and 4 is 12. Multiply every term in the equation by 12.
02

Distribute and simplify

Distribute 12 to each fraction in the equation: \[12 \times \frac{1}{3} w + 12 \times \frac{5}{4} = 12 \times w - 12 \times \frac{1}{4} \] Simplifying, this becomes: \[4w + 15 = 12w - 3\text{.} \]
03

Isolate the variable term

To isolate the variable term, move all terms with the variable to one side. Subtract 4w from both sides: \[ 4w + 15 - 4w = 12w - 4w - 3 \] Simplifying, this becomes: \[ 15 = 8w - 3 \]
04

Solve for the variable

To solve for the variable, isolate the 'w' term. First, add 3 to both sides: \[ 15 + 3 = 8w - 3 + 3 \] Simplifying, this becomes: \[ 18 = 8w\text{.} \] Finally, divide both sides by 8: \[ \frac{18}{8} = w \] Simplify: \[ w = \frac{9}{4} \]

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Key Concepts

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

Fraction Coefficients
When solving equations with fraction coefficients, you are dealing with variables that have fractions as multipliers. These can be tricky because fractions make arithmetic operations a bit more challenging.

For example, in the given problem: \(\frac{1}{3} w + \frac{5}{4} = w - \frac{1}{4}\), the coefficients of \(w\) and constant terms are fractions. To make calculations easier, eliminating these fractions by converting them to whole numbers can simplify the solving process.
Least Common Multiple
The least common multiple (LCM) is critical when you need to eliminate fractions from an equation. The LCM of two or more numbers is the smallest number that is a multiple of each number.

In this problem, the denominators are 3 and 4. The LCM of 3 and 4 is 12. By multiplying every term in the equation by 12, you can get rid of the denominators:
\[12 \times \frac{1}{3} w + 12 \times \frac{5}{4} = 12 \times w - 12 \times \frac{1}{4} \right.\]
This simplifies to:
\[4w + 15 = 12w - 3\]
Isolating Variables
Isolating the variable means getting the variable by itself on one side of the equation. This step is essential to solve for the variable.

In the equation \(4w + 15 = 12w - 3\), move all terms involving the variable \(w\) to one side and the constants to the other side. Subtract 4w from both sides:
\[4w + 15 - 4w = 12w - 4w - 3 \right.\]
Simplify this to: \[15 = 8w - 3\]
Then, to isolate \(w\), add 3 to both sides:
\[15 + 3 = 8w - 3 + 3 \right.\]
Simplify this to: \[18 = 8w\].
Simplifying Equations
Simplifying is the final step in solving the equation. It involves performing basic arithmetic to get the variable alone.

For example, after isolating the variable in the equation \[18 = 8w\], you solve for \(w\) by dividing both sides of the equation by 8:
\[\frac{18}{8} = w\]
Simplify the fraction to get:
\[w = \frac{9}{4}\].
Simplifying ensures that the solution is in its simplest form, making it easier to interpret.

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Most popular questions from this chapter

Solve each inequality, graph the solution on the number line, and write the solution in interval notation. $$v+12>3$$

Shopping The number of items, \(n\), a shopper can have in the express check-out lane is at most \(8 .\) Write this as an inequality.

Graph each inequality on the number line and write in interval notation. (a) \(\quad x \leq 5\) (b) \(\quad x \geq-1.5\) (c) \(x<-\frac{7}{3}\)

Use the formula \(A=\frac{1}{2} b h\). Solve for \(b\) (a) when \(A=65\) and \(h=13\) (b) in general

Use the formula \(d=r t\). Solve for \(t\) (a) when \(d=510\) and \(r=60\) (b) in general
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