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Chapter 5: Problem 7

Find the value of \(x\) in each proportion. a) \(\frac{x-3}{8}=\frac{x+3}{24}\) b) \(\frac{x+1}{6}=\frac{4 x-1}{18}\)

Short Answer

Expert verified
a) \(x = 6\); b) \(x = 4\)

Step by step solution

01

Cross Multiply for Part (a)

To solve the proportion \(\frac{x-3}{8}=\frac{x+3}{24}\), we start by cross-multiplying: \( (x - 3) \times 24 = (x + 3) \times 8 \). This gives us the equation \( 24(x - 3) = 8(x + 3) \).
02

Distribute and Simplify for Part (a)

Distribute each side: \( 24x - 72 = 8x + 24 \). Next, we want to get all terms involving \(x\) on one side and constant terms on the other, so subtract \(8x\) from both sides: \( 24x - 8x - 72 = 24 \). Simplify to get \( 16x - 72 = 24 \).
03

Solve for x in Part (a)

Add 72 to both sides to isolate the \(x\)-term: \( 16x = 96 \). Finally, divide both sides by 16 to solve for \(x\): \( x = 6 \).
04

Cross Multiply for Part (b)

Similarly, solve the proportion \(\frac{x+1}{6}=\frac{4x-1}{18}\) by cross multiplying: \( (x + 1) \times 18 = (4x - 1) \times 6 \). This results in the equation \( 18(x + 1) = 6(4x - 1) \).
05

Distribute and Simplify for Part (b)

Distribute to simplify: \( 18x + 18 = 24x - 6 \). Now, subtract \(18x\) from both sides to begin isolating \(x\): \( 18 = 6x - 6 \).
06

Solve for x in Part (b)

Add 6 to both sides of the equation to simplify: \( 24 = 6x \). Finally, divide both sides by 6 to solve for \(x\): \( x = 4 \).

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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 useful method when dealing with proportions. It helps us find an unknown value by transforming the equation. In our problem, we have proportions like \( \frac{x-3}{8} = \frac{x+3}{24} \). Here, cross-multiplication helps us by creating a single, simpler equation.
To cross-multiply, we multiply the numerator of one fraction by the denominator of the other. For the proportion above, this means:
  • Multiply \( (x - 3) \) by 24.
  • Multiply \( (x + 3) \) by 8.
This gives us the equation \( 24(x - 3) = 8(x + 3) \). Now, we have an equation without fractions. Cross-multiplication is super handy for simplifying these types of problems.
Solving Equations
Once you have simplified the equation using cross-multiplication, the next step is solving this equation. The goal is to find the value of the unknown variable, in this case, \( x \). Let's break it down into simple steps.
Using part (a) of our original problem, where we have \( 24(x - 3) = 8(x + 3) \), we start by distributing the multiplication to each term inside the parentheses:
  • 24 times \( x \) minus 24 times 3 gives \( 24x - 72 \).
  • 8 times \( x \) plus 8 times 3 gives \( 8x + 24 \).
Next, rearrange the equation to bring terms with \( x \) on one side and numbers on the other. This means subtracting 8x and adding 72 on both sides in this specific problem. The goal is always to make it easier to isolate \( x \).
Isolating Variables
After setting up your equation and simplifying where possible, it's time to isolate the variable. Isolating \( x \) involves getting \( x \) by itself on one side of the equation.
Using our simplified equation from part (a), \( 16x - 72 = 24 \), we add 72 to both sides to remove the constant term from the left side:
  • This leads to \( 16x = 96 \).
The final step is to isolate \( x \). This is done by dividing every term by the coefficient of \( x \) (which is 16 in this case):
  • Divide both sides by 16, resulting in \( x = 6 \).
Always remember, isolating the variable means ensuring \( x \) is all by itself on one side of the equation, making it straightforward to solve for its value.

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Find the value of \(x\) in each proportion. a) \(\frac{x+1}{x}=\frac{x}{x-1}\) b) \(\frac{x+2}{x}=\frac{2 x}{x-2}\)

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