Problem 18 - Algebra 1: Concepts and Skills

Question

Solve the proportion using the cross product property. Check your solution. $$ \frac{42}{28}=\frac{3}{x} $$

Step-by-Step Solution

Answer

The solution to the proportion is \(x = 2\).

Step 1: Apply the cross product property

The cross product property can be applied to the provided proportion, which gives us \(42 * x = 28 * 3\).

Step 2: Simplify the equation

The equation from step 1 can be simplified to \(42x = 84\).

Step 3: Solve for 'x'

Solving for 'x' involves dividing both sides of the equation by 42. This results in \(x = 2\).

Step 4: Check your solution

Checking the solution involves substituting \(x = 2\) back into the original proportion to see if it holds true. Substitution gives us \(\frac{42}{28} = \frac{3}{2}\), which simplifies to \(\frac{3}{2} = \frac{3}{2}\). Because this statement is true, the solution is confirmed as correct.

Key Concepts

Cross Product Property

When dealing with proportions, the cross product property is a powerful tool that makes solving them straightforward. A proportion is essentially an equation stating that two ratios are equivalent. In mathematical terms, if \( \frac{a}{b} = \frac{c}{d} \), we can use the cross product property. This property allows us to convert the proportion into an equation without fractions by cross-multiplying, which means multiplying the numerator of one fraction by the denominator of the other. Here’s the simple breakdown:

• Multiply the numerator of the first ratio by the denominator of the second ratio.
• Do the same for the numerator of the second ratio and the denominator of the first ratio.

This gives us: \( a \cdot d = b \cdot c \).
In the original exercise, you apply this property to \( \frac{42}{28} = \frac{3}{x} \), resulting in \( 42 \cdot x = 28 \cdot 3 \). Doing so helps eliminate the fractions, making it easier to solve for the unknown value.

Simplifying Equations

Once you have successfully applied the cross product property, the next step is to simplify the resulting equation. Simplifying an equation means making it more manageable and easier to solve. After using the cross product property, you'll often be left with a straightforward linear equation.
With our example, we derived the equation \( 42x = 84 \). Simplification here involves isolating the variable \( x \). Since \( x \) is being multiplied by 42, you will need to do the opposite operation to simplify it, which is division.

• Divide both sides of the equation by the same non-zero number.
• In the example, you divide by 42 to simplify \( 42x = 84 \).

This results in \( x = 2 \).
This process not only brings you closer to solving for the unknown, but it also ensures that the equation maintains its balance.

Checking Solutions

Checking your solution is a critical step in solving equations as it ensures accuracy. It involves verifying that your solution satisfies the original equation or proportion. By substituting the found solution back into the original proportion, you can confirm its correctness.
In the exercise, after solving \( x = 2 \), you should replace \( x \) in the original proportion \( \frac{42}{28} = \frac{3}{x} \) to verify:

• Substitute \( x = 2 \) in place of \( x \).
• This gives \( \frac{42}{28} = \frac{3}{2} \).
• Simplify both sides to check if they are equivalent: \( \frac{3}{2} = \frac{3}{2} \).

If both sides match, your solution is correct. If they don't, you'll need to re-evaluate your steps in solving the equation. Checking avoids errors and solidifies your understanding of solving proportions.