Problem 65 - Algebra 2

Question

PREREQUISITE SKILL Use cross products to solve each proportion. $$ \frac{x}{5}=\frac{x+1}{8} $$

Step-by-Step Solution

Answer

The solution to the proportion is \( x = \frac{5}{3} \).

Step 1: Understand Cross Products

The cross product of a proportion involves multiplying the numerator of one fraction by the denominator of the other fraction, equating the products from both sides of the equation. For the given proportion \( \frac{x}{5} = \frac{x+1}{8} \), we will find the cross products to solve for \( x \).

Step 2: Set Up the Cross Products

To find the cross products, multiply the numerator of the first fraction (\( x \)) with the denominator of the second fraction (\( 8 \)); likewise, multiply the numerator of the second fraction (\( x+1 \)) with the denominator of the first fraction (\( 5 \)). We have:\[ 8x = 5(x+1) \]

Step 3: Simplify the Equation

Distribute \( 5 \) on the right side of the equation:\[ 8x = 5x + 5 \]

Step 4: Isolate the Variable

Subtract \( 5x \) from both sides to get the terms involving \( x \) on one side:\[ 8x - 5x = 5 \]This simplifies to:\[ 3x = 5 \]

Step 5: Solve for \( x \)

Divide both sides by 3 to solve for \( x \):\[ x = \frac{5}{3} \]

Step 6: Verify the Solution

Substitute \( x = \frac{5}{3} \) back into the original proportion to verify:\[ \frac{5/3}{5} = \frac{(5/3)+1}{8} \]Simplify both fractions, which should be equal if the solution is correct.

Key Concepts

Proportions

A proportion is an equation that states that two ratios are equivalent. It's like saying two fractions represent the same part of a whole or that they are in balance. Proportions are common in various problems, especially those involving scaling or similarity. To solve a problem involving proportions, you need to understand this balance between two ratios.

• For example, if one side of a proportion is \( \frac{1}{2} \), it suggests that for every one part of something, there are two equal parts in total.
• Likewise, with \( \frac{2}{4} \), although numerically different, it represents the same relationship as \( \frac{1}{2} \).

This relationship allows us to use the principle of cross products to find unknown values.

Solving Equations

Solving equations is the process of finding the unknown value that makes the equation true. When you solve an equation, you are searching for the balance that exists within that mathematical statement. With proportions specifically, we often use cross multiplication.

• This involves taking the numerator of one fraction and multiplying it by the denominator of the other fraction, equating these products across the equation.
• For example, take \( \frac{a}{b} = \frac{c}{d} \); performing cross multiplication gives us \( ad = bc \).
• By setting these products equal, we're leveraging the property of equality to find a value that satisfies the equation.

This kind of setup simplifies the equation into a more manageable form, making it easier to isolate and solve for the unknown variable.

Algebraic Manipulation

Algebraic manipulation is key to solving equations involving proportions. Once you've cross multiplied, the resulting equation often needs to be simplified. This involves applying basic algebraic rules to isolate the unknown variable.

• For instance, consider the equation \( 8x = 5x + 5 \).
• You would subtract \( 5x \) from both sides to bring terms involving \( x \) together: \( 8x - 5x = 5 \).
• The next step is to perform the arithmetic operation which leads you to \( 3x = 5 \).
• Finally, divide by the coefficient of \( x \) to solve for it, resulting in \( x = \frac{5}{3} \).

These steps demonstrate how algebraic manipulation helps break down the problem, progressively honing in on the unknown variable until it's isolated and resolved.