Problem 6 - Algebra A Combined Function

Question

Solve each proportion. $$ \frac{x+1}{x+2}=\frac{5}{3} $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{7}{2}\).

Step 1: Understanding the Proportion

In this problem, the expression \( \frac{x+1}{x+2} = \frac{5}{3} \) is a proportion. A proportion asserts that two ratios are equal. Our aim is to find the value of \( x \) that makes this true.

Step 2: Cross-Multiplication

To solve the proportion \( \frac{x+1}{x+2} = \frac{5}{3} \), use the cross-multiplication method. Multiply the numerator of the first fraction by the denominator of the second fraction, and multiply the numerator of the second fraction by the denominator of the first. This gives: \[(x+1) \cdot 3 = (x+2) \cdot 5\]

Step 3: Simplify the Equation

Simplify both sides of the equation obtained from cross-multiplication:Left-side: \[3(x + 1) = 3x + 3\] Right-side:\[5(x + 2) = 5x + 10\] The equation now is: \[3x + 3 = 5x + 10\]

Step 4: Isolate the Variable

Subtract \(3x\) from both sides of the equation to bring \(x\) terms to one side:\[3 = 2x + 10\]Next, subtract \(10\) from both sides to isolate terms involving \(x\):\[3 - 10 = 2x\]This simplifies to:\[-7 = 2x\]

Step 5: Solve for x

Divide both sides of the equation by \(2\) to get:\[x = \frac{-7}{2}\]

Step 6: Check Your Solution

Substitute \(x = -\frac{7}{2}\) back into the original proportion to check correctness.Original Proportion: \[\frac{x+1}{x+2} = \frac{5}{3}\] Substitute: \[\frac{-\frac{7}{2} + 1}{-\frac{7}{2} + 2} = \frac{5}{3}\]This simplifies to: \[\frac{-\frac{5}{2}}{-\frac{3}{2}} = \frac{5}{3}\]After simplification, both sides are equal, confirming \(x = -\frac{7}{2}\).

Key Concepts

Cross-Multiplication

One effective method to solve proportion problems, like the one given, is cross-multiplication. Cross-multiplication is a mathematical technique used to eliminate the fractions in a proportion and create a straightforward equation. This makes solving for the unknown variable easier. To perform cross-multiplication, you'll multiply the numerator of each fraction by the denominator of the opposite fraction. For instance, in the equation \( \frac{x+1}{x+2} = \frac{5}{3} \), you multiply \( (x+1) \) by \( 3 \), and \( 5 \) by \( (x+2) \). This results in an equation without fractions:

• The left side becomes: \( 3(x + 1) \)
• The right side becomes: \( 5(x + 2) \)

Thus, the new equation is \( 3(x + 1) = 5(x + 2) \). This equation is easier to manage and is the first step toward finding the value of \( x \). Cross-multiplying is useful as it simplifies the equation to a form where you can apply algebraic techniques to find unknowns.

Isolation of Variables

After using cross-multiplication, we need to isolate the variable to solve for it. In our example, we have an equation \( 3x + 3 = 5x + 10 \). The goal is to get all terms involving \( x \) on one side of the equation and constants on the other.Begin by subtracting \( 3x \) from both sides to ensure that \( x \) is present only on one side of the equation. This gives us \( 3 = 2x + 10 \). Next, subtract \( 10 \) from both sides to further isolate terms involving \( x \):

• The equation simplifies to \( -7 = 2x \).

Finally, to fully isolate \( x \), divide both sides by \( 2 \), leading to \( x = \frac{-7}{2} \).Tips for isolating variables:

• Always perform the same operation on both sides of the equation.
• Simplify each step to reduce potential mistakes.

This systematic approach helps to discover the value of \( x \), turning a complex problem into a much simpler one.

Checking Solutions

It's vital to check your solution after you've calculated it to ensure accuracy, especially in proportion problems. When you solve for \( x \), you get \( x = \frac{-7}{2} \). To verify this solution, substitute \( x \) back into the original proportion:\( \frac{x+1}{x+2} = \frac{5}{3} \)Replace \( x \) with \( -\frac{7}{2} \):

• For the numerator: \( -\frac{7}{2} + 1 \), which simplifies to \( -\frac{5}{2} \)
• For the denominator: \( -\frac{7}{2} + 2 \), which simplifies to \( -\frac{3}{2} \)

Thus, the fraction becomes \( \frac{-\frac{5}{2}}{-\frac{3}{2}} \).Simplifying both numerator and denominator cancels out the negatives and gives the fraction \( \frac{5}{3} \), which matches the right side of the original equation.Reasons to check solutions:

• Confirms the accuracy of your solution.
• Ensures that no steps were missed during calculation.

Checking your solution is a final yet crucial safeguard against errors—making sure everything balances out and is correct.