Problem 13 - Algebra 2 and Trigonometry

Question

Solve each proportion for the variable. \(\frac{a-2}{2 a}=\frac{5}{14}\)

Step-by-Step Solution

Answer

The solution is \( a = 7 \).

Step 1: Cross Multiply

To solve the proportion \( \frac{a-2}{2a} = \frac{5}{14} \), start by cross-multiplying. This means you multiply the numerator of one fraction by the denominator of the other fraction and set this equal to the numerator of the second fraction multiplied by the denominator of the first fraction. Thus, you get:\[ (a - 2) imes 14 = 5 imes 2a \]This simplifies to:\[ 14(a - 2) = 10a \]

Step 2: Distribute and Simplify

Next, distribute the 14 on the left side of the equation:\[ 14a - 28 = 10a \]Now, the equation is simplified to use standard subtraction and addition.

Step 3: Isolate the Variable

To isolate \( a \), bring all terms involving \( a \) to one side of the equation. Subtract \( 10a \) from both sides:\[ 14a - 10a = 28 \]This simplifies to:\[ 4a = 28 \]

Step 4: Solve for the Variable

Now, solve for \( a \) by dividing both sides of the equation by 4:\[ a = \frac{28}{4} \]This simplifies to:\[ a = 7 \]

Step 5: Verify the Solution

To ensure the solution is correct, substitute \( a = 7 \) back into the original proportion:\[ \frac{7-2}{2\times7} = \frac{5}{14} \]Simplify the left side:\[ \frac{5}{14} = \frac{5}{14} \]Since both sides are equal, the solution \( a = 7 \) is verified.

Key Concepts

Cross Multiplication

Cross multiplication is a powerful method used to solve proportions, like the one given in this exercise: \( \frac{a-2}{2a} = \frac{5}{14} \). This technique helps simplify the equation by eliminating fractions. To apply it, you multiply the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction. For the given problem, this results in:

• \( (a - 2) \times 14 = 5 \times 2a \)

This sets up a basic equation without fractions and isolates separate parts of the equation on each side. By making sure both sides of the equation are equal, you can work towards finding the variable.

Simplifying Equations

After applying cross multiplication, the next step is simplifying the equation. This involves dealing with expressions like \(14(a - 2) = 10a\). At this point:

• The distributive property can be used to expand \(14(a - 2)\).
• Distribute 14 to both \(a\) and \(-2\), resulting in \(14a - 28\).

Once distribution is complete, the equation \(14a - 28 = 10a\) is a tool for further simplification. The goal is to have a clearer path to continue working towards isolating the variable by operating on both sides of the equation.

Distributive Property

The distributive property is essential in progressing with the solution of equations like \(14(a - 2) = 10a\). It's used to distribute one term across terms inside parentheses:

• Multiply 14 by \(a\) to get \(14a\).
• Multiply 14 by \(-2\) to get \(-28\).

This simplification doesn't yield a solution directly, but it refines the equation to \(14a - 28 = 10a\). This clearer view prepares you for the next step, where you aim to bring all terms involving \(a\) together to help solve for this variable efficiently.

Isolating the Variable

Isolating the variable—a crucial step in solving equations—means rearranging the equation so that you have the variable by itself on one side. With our example \(14a - 28 = 10a\), it involves:

• Subtracting \(10a\) from both sides to group all \(a\) terms, resulting in \(4a = 28\).
• Simple arithmetic operations make \(a\) more manageable, through division or multiplication.

Once the equation is reduced to \(4a = 28\), divide each side by 4 to isolate \(a\), bringing you to \(a = 7\). Verification with the original proportion ensures this is the correct solution, confirming our process was sound.