Problem 22 - Algebra 1

Question

Solve the proportion. Check for extraneous solutions. $$\frac{24}{5}=\frac{9}{y+2}$$

Step-by-Step Solution

Answer

The solution to the proportion is y = -1/8 and there are no extraneous solutions.

Step 1: Cross Multiply

Cross multiply the equations: 24*(y+2) = 9*5. This reduces to 24y + 48 = 45.

Step 2: Solve for the Variable

To isolate 'y' subtract 48 from each side of the equation: 24y = -3. Then, divide each side by 24 to isolate y: y = -3/24 = -1/8.

Step 3: Check for Extraneous Solutions

Substitute y = -1/8 into the original equation to check for extraneous solutions. The equation becomes 24/5 = 9/(-1/8+2). After simplification, we get 24/5 = 9/(15/8). This simplifies too 24/5 = 48/15. When further simplified, we get 24/5 = 24/5 which proves that y = -1/8 is the correct solution of the proportion and there are no extraneous solutions.

Key Concepts

Cross Multiplication

Cross multiplication is a simple yet powerful method used to solve equations involving proportions. A proportion involves two ratios that are set equal to each other. When you cross-multiply, you multiply the numerator of one ratio by the denominator of the other ratio and vice versa. This method helps in simplifying the equation so that you can solve for the unknown variable more easily.

Consider the proportion \( \frac{24}{5} = \frac{9}{y+2} \). With cross multiplication, you multiply the numerator of the first ratio (24) with the denominator of the second ratio (\(y+2\)), and the denominator of the first ratio (5) with the numerator of the second ratio (9), giving us the equation:

• 24 * (y + 2) = 9 * 5

This results in the equation:

24y + 48 = 45.

Cross multiplication is particularly useful because it helps eliminate the fraction, making it easier to solve the remaining equation.

Extraneous Solutions

Extraneous solutions occur when you perform certain mathematical operations that introduce solutions that do not actually satisfy the original equation. It commonly happens when dealing with equations involving variables in the denominators, like in proportions.

After performing all necessary computations, it is crucial to substitute the solution back into the original equation to verify that it is valid.

Following from the example, we solved for \( y \) and found \( y = -\frac{1}{8} \). We then substitute it back into the original proportion \( \frac{24}{5} = \frac{9}{y+2} \):

• \( \frac{24}{5} = \frac{9}{-\frac{1}{8}+2} \)
• This simplifies to \( \frac{24}{5} = \frac{9}{\frac{15}{8}} \)
• Further simplifying gives \( \frac{24}{5} = \frac{48}{15} \)
• Upon simplification, we see that \( \frac{24}{5} \) equals \( \frac{24}{5} \).

Since the original equation holds true, we confirm that \( y = -\frac{1}{8} \) is not an extraneous solution, affirming its validity.

Solving Equations

Solving equations involves isolating the unknown variable on one side to find its value. This process includes performing operations such as addition, subtraction, multiplication, or division on both sides of the equation, ensuring the equation remains balanced.

Continuing from the cross multiplication step, you have the equation 24y + 48 = 45 after cross-multiplying. The next step is to solve for \( y \):

• Subtract 48 from both sides: 24y = -3
• Divide both sides by 24: \( y = -\frac{3}{24} \)

This simplifies to \( y = -\frac{1}{8} \) after dividing the numerator and the denominator by 3.

Solving equations is like making a balanced seesaw even; what you do to one side, you must do to the other. In complex problems, breaking down each step keeps things simple and solves the equation efficiently.