Problem 19 - Algebra 1

Question

Solve the proportion. Check for extraneous solutions. $$\frac{5}{8}=\frac{c}{9}$$

Step-by-Step Solution

Answer

The solution to the proportion is \( c = \frac{45}{8} \). There are no extraneous solutions.

Step 1: Setup Proportion

Start with the given proportion and clearly write it down. This is what we're asked to solve: \( \frac{5}{8}=\frac{c}{9} \)

Step 2: Cross Multiply

Cross multiply to get rid of the fractions. This involves multiplying diagonally across the equal sign. Multiply 5 and 9 together and 8 and c together: \( 5 \times 9 = 8 \times c \)

Step 3: Simplify

Simplify the multiplication. \( 45 = 8c \)

Step 4: Isolate c

To find the value of c, divide both sides of the equation by the coefficient of c, which is 8 in order to isolate c: \( \frac{45}{8} = c \)

Step 5: Check for Extraneous Solutions

Substitute the value of c back into the original equation and check if both sides of the equation remains equal. In this case, the solution is \( \frac{45}{8} \), and it's not an extraneous solution, because it makes the original equation true: \( \frac{5}{8}=\frac{45/8}{9} \) or \( \frac{5}{8}=\frac{5}{8} \)

Key Concepts

Cross Multiplication

Cross multiplication is a straightforward method used to solve proportions. A proportion involves two fractions set equal to each other, and cross multiplication helps eliminate the fractions, allowing us to solve for the unknown variable more easily.

You perform cross multiplication by multiplying the numerator of each fraction by the denominator of the opposite fraction:

• For the proportion \(\frac{5}{8} = \frac{c}{9}\), multiply the numerator of the first fraction (5) by the denominator of the second fraction (9), resulting in \(5 \times 9\).
• Then multiply the denominator of the first fraction (8) by the numerator of the second fraction (c), resulting in \(8 \times c\).

After cross multiplying, you get an equation without fractions: \(45 = 8c\). Cross multiplication is efficient because it quickly sets up a simple equation for you to solve, bringing you closer to finding the value of the unknown variable.

Equation Solving

Once cross multiplication gives us an equation without fractions, the next goal is to solve for the unknown variable. Typically, this requires isolating the variable on one side of the equation.

In our example, we end up with the equation \(45 = 8c\). To find \(c\), we need to get \(c\) by itself on one side of the equation. This is done by dividing both sides by the coefficient of \(c\) (which is 8 in this case):

• \(\frac{45}{8} = c\)

This process of isolating the variable involves straightforward arithmetic operations that maintain the balance of the equation. It's essential because it allows us to determine the precise value of the unknown, which is \(\frac{45}{8}\) here, ensuring that the proportion is correct.

Extraneous Solutions

Extraneous solutions refer to results that emerge in the solving process but do not satisfy the original equation. These sometimes appear due to operations that might introduce false or irrelevant solutions.

In solving proportions, always substitute your solution back into the original equation to verify accuracy:

• Insert \(c = \frac{45}{8}\) back into the original proportion \(\frac{5}{8} = \frac{c}{9}\).
• Check if both sides are equal: \(\frac{5}{8} = \frac{45/8}{9} = \frac{5}{8}\).

Here, the original equation holds true, confirming that \(\frac{45}{8}\) is not extraneous. This final check is crucial to ensure that your solution genuinely works and isn’t simply a byproduct of the solving process.