Problem 34 - Elementary Algebra

Question

Solve each of the equations. $$\frac{4(x+3)}{7}=\frac{2(x-6)}{5}$$

Step-by-Step Solution

Answer

The solution is \( x = -24 \).

Step 1: Cross-Multiply

To eliminate the fractions, cross-multiply: Multiply the numerator of the first fraction by the denominator of the second and the numerator of the second fraction by the denominator of the first.This gives us:\[ 4(x+3) imes 5 = 2(x-6) imes 7 \]Simplify to:\[ 20(x + 3) = 14(x - 6) \]This equation no longer has any fractions and is easier to solve.

Step 2: Distribute the Constants

Distribute the constants 20 and 14 across the parentheses on both sides of the equation:\[ 20x + 60 = 14x - 84 \]

Step 3: Move x Terms to One Side

To isolate the terms with \( x \) on one side of the equation, subtract \( 14x \) from both sides:\[ 20x - 14x + 60 = -84 \]Simplifying gives:\[ 6x + 60 = -84 \]

Step 4: Move Constant Terms to the Other Side

Subtract 60 from both sides to get the equation ready for isolating \( x \):\[ 6x = -84 - 60 \]Simplifying gives:\[ 6x = -144 \]

Step 5: Solve for x

Divide both sides by 6 to solve for \( x \):\[ x = \frac{-144}{6} \]Simplifying gives:\[ x = -24 \]

Step 6: Verify the Solution

Substitute \( x = -24 \) back into the original equation to verify:\[ \frac{4(-24 + 3)}{7} = \frac{2(-24 - 6)}{5} \]Calculate each side:Left side: \( \frac{4(-21)}{7} = -12 \).Right side: \( \frac{2(-30)}{5} = -12 \).Both sides are equal, confirming \( x = -24 \) is the correct solution.

Key Concepts

Cross-Multiplication

Cross-multiplication is an effective technique used to solve equations involving fractions. It allows us to eliminate the fractions and obtain an equation that is often simpler to solve. Cross-multiplication works by multiplying the numerator of each fraction by the denominator of the other fraction. Let's illustrate this with the equation provided, \( \frac{4(x+3)}{7} = \frac{2(x-6)}{5} \).

• First, identify the numerator and denominator of the fractions.
• Multiply the numerator of the first fraction by the denominator of the second fraction, yielding \( 4(x+3) \times 5 \).
• Similarly, multiply the numerator of the second fraction by the denominator of the first fraction, resulting in \( 2(x-6) \times 7 \).

The cross-multiplication gives us an equation without fractions: \( 20(x+3) = 14(x-6) \). This step simplifies the process of solving the equation by transforming it into a linear equation.

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to simplify expressions. It involves multiplying each term inside a set of parentheses by a factor outside the parentheses. In the equation \( 20(x+3) = 14(x-6) \), using the distributive property helps to expand and simplify both sides.

• On the left side, distribute the \( 20 \) across \( x+3 \). This gives: \( 20 \times x + 20 \times 3 = 20x + 60 \).
• On the right side, distribute the \( 14 \) across \( x-6 \), resulting in \( 14 \times x - 14 \times 6 = 14x - 84 \).

This transformation is essential in solving linear equations as it clears the way to isolate variables and proceed with further simplification. Once completed, the equation reads \( 20x + 60 = 14x - 84 \), making it easier to solve for the variable \( x \).

Verifying Solutions

Verifying solutions is a crucial step after solving an equation to ensure that the solution is correct. It involves substituting the solution back into the original equation and checking if both sides are equal. For our original equation \( \frac{4(x+3)}{7} = \frac{2(x-6)}{5} \), we've solved for \( x = -24 \).

• First, substitute \( x = -24 \) into the left-hand side of the equation: \( \frac{4(-24 + 3)}{7} = \frac{4(-21)}{7} = -12 \).
• Next, substitute \( x = -24 \) into the right-hand side of the equation: \( \frac{2(-24 - 6)}{5} = \frac{2(-30)}{5} = -12 \).

Both sides of the equation equal \( -12 \), confirming that our solution \( x = -24 \) is correct. Verification assures that the solution satisfies the original equation, providing a final check to avoid errors.