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Chapter 6: Problem 69

Solve the proportions. $$\frac{x+3}{x}=\frac{5}{4}$$

Short Answer

Expert verified
x = 12

Step by step solution

01

- Set up the equation

Start by writing the proportion equation: \[ \frac{x+3}{x} = \frac{5}{4} \]
02

- Cross-multiply to eliminate the fraction

Cross-multiplying the two fractions will help eliminate the denominators: \[ 4(x+3) = 5x \]
03

- Expand the equation

Expand the left side of the equation: \[ 4x + 12 = 5x \]
04

- Isolate the variable

Subtract 4x from both sides of the equation to isolate the variable: \[ 12 = 5x - 4x \] \[ 12 = x \]
05

- Verify the solution

Substitute x = 12 back into the original equation to ensure the solution is correct: \[ \frac{12 + 3}{12} = \frac{15}{12} = \frac{5}{4} \], which confirms the solution.

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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 eliminate the fractions in an equation by multiplying the numerator of each fraction by the denominator of the other fraction. In our exercise, we have the proportion \[ \frac{x+3}{x} = \frac{5}{4} \]. To start, we cross-multiply the two fractions. This means multiplying the left numerator (\(x+3\)) by the right denominator (4) and the left denominator (x) by the right numerator (5). The result is: \[ 4(x+3) = 5x \]. This step helps us to remove the fractions and simplifies the process of solving for x.
Isolating Variables
Next, we need to isolate the variable x to solve the equation. After cross-multiplication, our equation is \[ 4(x+3) = 5x \], which expands to \[ 4x + 12 = 5x \]. To isolate x, we subtract 4x from both sides: \[ 4x + 12 - 4x = 5x - 4x \]. This simplifies to \[ 12 = x \]. By moving all terms with the variable to one side and the constants to the other, we effectively isolate x, making it easy to see that x equals 12. This step is crucial in solving equations as it allows us to determine the value of the unknown variable.
Verification of Solutions
After solving for the variable, it鈥檚 important to verify our solution to ensure it is correct. We do this by substituting the value of x back into the original equation. For our example, substituting x = 12 gives us: \[ \frac{12+3}{12} = \frac{15}{12} = \frac{5}{4} \]. Since the left-hand side equals the right-hand side of the original proportion, our solution is verified. Verification is an essential step as it confirms that the solutions are accurate and the equation balances correctly.

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