Problem 28 - Intermediate Algebra

Question

Solve each proportion. $$ \frac{a}{a+1}=\frac{a+2}{a} $$

Step-by-Step Solution

Answer

The solution for \( a \) is \( -\frac{2}{3} \).

Step 1: Cross-Multiply the Proportions

The first step to solving the proportion \( \frac{a}{a+1} = \frac{a+2}{a} \) is to cross-multiply. Cross-multiplying involves multiplying the numerator of each fraction by the denominator of the other fraction. This gives us the equation: \( a \cdot a = (a+2)(a+1) \).

Step 2: Expand Both Sides of the Equation

Now, let's expand both sides of the equation from Step 1. The left side is \( a^2 \). For the right side, use the distributive property: \((a+2)(a+1) = a(a+1) + 2(a+1) = a^2 + a + 2a + 2 = a^2 + 3a + 2\). Now, the equation is \( a^2 = a^2 + 3a + 2 \).

Step 3: Simplify the Equation

Subtract \( a^2 \) from both sides to simplify the equation: \( a^2 - a^2 = a^2 + 3a + 2 - a^2 \), which results in \( 0 = 3a + 2 \).

Step 4: Solve for 'a'

Next, isolate \( a \) by subtracting 2 from both sides: \( -2 = 3a \). Divide both sides by 3 to solve for \( a \): \( a = -\frac{2}{3} \).

Key Concepts

Cross-Multiplication

Cross-multiplication is a technique used to solve equations involving proportions, where two fractions are set equal to each other. When you cross-multiply, you multiply the numerator of each fraction by the denominator of the other fraction. The primary goal here is to eliminate the fractions to simplify the equation. Here's how you can apply cross-multiplication to solve a proportion:

• Take the two fractions, for example, \( \frac{a}{b} = \frac{c}{d} \).
• Multiply \( a \) by \( d \), and \( b \) by \( c \).
• This results in the equation: \( a \times d = b \times c \).

From here, you end up with a basic equation that's often easier to manipulate and solve. In the original exercise, cross-multiplying transformed \( \frac{a}{a+1} = \frac{a+2}{a} \) into \( a \cdot a = (a+2)(a+1) \), getting rid of the fractions to allow direct expansion in the next step.

Distributive Property

The distributive property is a useful algebraic principle that allows you to expand expressions in the form \( (x + y)(z) \). This property states that you can distribute the multiplication over each addition within parentheses. In mathematical terms, the distributive property is written as:

• \( x(y + z) = xy + xz \)
• This means you multiply \( x \) by \( y \) and \( z \) separately, and then add the products.

Applying this in the solution, you expand \((a+2)(a+1)\) as follows:

• First, \( a \times (a + 1) \) yields \( a^2 + a \).
• Then, \( 2 \times (a + 1) \) gives \( 2a + 2 \).
• Add those results together to get: \( a^2 + a + 2a + 2 = a^2 + 3a + 2 \).

The distributive property allowed us to break down the expression into simpler parts, setting up the equation for further simplification.

Solving Equations

Solving equations is the process of finding the variable value that satisfies the equation. In algebra, it typically involves isolating the variable on one side of the equation. When you solve equations, the goal is to simplify the equation step-by-step until it becomes manageable. Here's the approach used in the solution:

• After cross-multiplying, we simplified the equation to \( a^2 = a^2 + 3a + 2 \).
• Next, subtract \( a^2 \) from both sides, which results in a simpler linear equation: \( 0 = 3a + 2 \).
• To isolate \( a \), subtract 2 from both sides: \( -2 = 3a \).
• Finally, divide by 3 to solve for \( a \): \( a = -\frac{2}{3} \).

This sequence of operations helps identify \( a \) as \( -\frac{2}{3} \), as calculated in the original exercise. Always check your solution by plugging it back into the original equation to verify correctness.