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Chapter 4: Problem 41

Solve the equations and inequalities. $$\frac{w+3}{4}=\frac{w+4}{3}$$

Short Answer

Expert verified
The value of \(w\) is \(-7\).

Step by step solution

01

Understand the Problem

We need to solve the equation \(\frac{w+3}{4}=\frac{w+4}{3}\). This means finding the value of \(w\) that makes both sides equal.
02

Cross-Multiply to Eliminate the Fractions

To eliminate the fractions, cross-multiply. This means \(4 \cdot (w+4) = 3 \cdot (w+3)\). This simplifies to: \[4(w+4) = 3(w+3)\]
03

Distribute and Simplify

Distribute the numbers through the parentheses: \[4w + 16 = 3w + 9\]
04

Isolate the Variable

Subtract 3w from both sides to get the variable terms on one side: \[4w - 3w + 16 = 9\]. This simplifies to: \[w + 16 = 9\]
05

Solve for \(w\)

Subtract 16 from both sides to isolate \(w\): \[w + 16 - 16 = 9 - 16\]. This simplifies to: \[w = -7\]

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Key Concepts

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

Cross-Multiplication
Cross-multiplication is a method used to solve equations involving fractions. Essentially, you multiply the numerator of one fraction by the denominator of the other fraction, and vice versa. This helps eliminate the fractions, making the equation simpler to work with.
Let's apply it to our equation \(\frac{w+3}{4}=\frac{w+4}{3}\):
  • You cross-multiply: \(4 \times (w + 4)\) and \(3 \times (w + 3)\).
  • So, now you have: \(4(w + 4) = 3(w + 3)\).
By doing this, we no longer have fractions in our equation, making it easier to solve.
Distributive Property
The distributive property is used when you need to multiply a single term by two or more terms inside a parenthesis. Essentially, you distribute the multiplication across each term inside the parenthesis.
In our problem, after cross-multiplying, we got the equation: \(4(w + 4) = 3(w + 3)\).
  • Using the distributive property, we multiply 4 by both \(w\) and 4, and 3 by both \(w\) and 3.
  • So, it becomes: \(4w + 16 = 3w + 9\).
This step breaks down the equation, making it simpler to isolate the variable in the next steps.
Isolate Variable
Isolating the variable means getting the variable by itself on one side of the equation. This often involves addition, subtraction, multiplication, or division.
In our example, after using the distributive property, we have: \(4w + 16 = 3w + 9\).
  • First, subtract \(3w\) from both sides to move the variable terms to one side: \(4w - 3w + 16 = 9\), which simplifies to \(w + 16 = 9\).
Next, move the constant term by subtracting 16 from both sides:
  • \(w + 16 - 16 = 9 - 16\)
  • This simplifies to \(w = -7\).
Now, the variable is isolated and we have found the value of \(w\).

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Most popular questions from this chapter

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. A chemist wants to make \(90 \mathrm{ml}\) of a \(40 \%\) hydrochloric acid solution. She also wants to use all of the \(30 \mathrm{ml}\) of \(20 \%\) hydrochloric acid she has on hand. What percent solution must she use for the remaining 60 \(\mathrm{ml} ?\)

If the exercise is an equation, solve it; if not, perform the indicated operations and express your answer as a single fraction. $$\frac{2 x-1}{5}-\frac{x-7}{3}$$

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. On a certain day the minimum temperature was \(14^{\circ} \mathrm{F},\) and the maximum temperature was \(23^{\circ} \mathrm{F}\). What was the temperature range in degrees Celsius? \(\left(\text { Use } F=\frac{9}{5} \mathrm{C}+32 .\right)\)

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{5}{4 a b^{2}}+\frac{9}{10 a^{3} b}$$

In each of the following exercises, perform the indicated operations. Express your answer as a single fraction reduced to lowest terms. $$\frac{7}{6 a^{2} b}+\frac{3}{4 a b^{3}}$$
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