Problem 16 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(\frac{3}{4}(x+6)=12\)

Step-by-Step Solution

Answer

The solution for the equation is \(x = 10\).

Step 1: Clear the Fraction

To make the equation easier to solve, multiply every term by 4 to clear the fraction: \(4 * \frac{3}{4}(x+6) = 4 * 12\), which simplifies to: \(3(x+6) = 48\).

Step 2: Distribute

Now distribute the multiplication across the bracket: \(3x + 18 = 48\).

Step 3: Subtract 18

Subtract 18 from both sides of the equation to get 'x' alone: \(3x = 48 - 18\), which simplifies to: \(3x = 30\).

Step 4: Solve for x

Finally, divide both sides by 3 to find 'x': \(x = 30 / 3\), which simplifies to: \(x = 10\).

Key Concepts

Fractions in Equations

Dealing with fractions in equations can seem daunting at first, but once you understand the process, it becomes much more manageable. In the equation \(\frac{3}{4}(x+6)=12\), the fraction \(\frac{3}{4}\) is multiplied by the entire expression \(x+6\). The first step in solving this is to "clear" the fraction, which makes the equation easier to work with.
Clearing a fraction involves multiplying every term by the denominator of the fraction—in this case, 4. By doing so, you remove the fraction and get a simpler expression. Multiply both sides of the equation by 4:

• The \(4\) cancels out the denominator \(4\) in \(\frac{3}{4}\), leaving you with \(3(x+6)\).
• The right-hand side becomes \(4 \times 12 = 48\).

Now, the equation \(\frac{3}{4}(x+6) = 12\) turns into \(3(x+6) = 48\), free of fractions and ready for the next steps.

Distributive Property

The distributive property is a useful tool for simplifying expressions and solving equations. It allows you to distribute multiplication over an addition or subtraction inside a parenthesis. In the equation \(3(x+6) = 48\), we must apply the distributive property. This means you multiply the number outside the parenthesis, which is 3, by each term inside the parenthesis, which are \(x\) and 6.
Using the distributive property here, multiply 3 by \(x\) and then 3 by 6:

• \(3 \times x = 3x\)
• \(3 \times 6 = 18\)

This gives you the new expression \(3x + 18 = 48\). It's a crucial step for breaking down complex expressions into simpler components that are easy to work with. After applying the distributive property, you're one step closer to isolating the variable \(x\).

Isolation of Variables

Isolating the variable is the main goal when solving linear equations. It involves getting the variable (here, \(x\)) all by itself on one side of the equation. After applying the distributive property, we have \(3x + 18 = 48\). To isolate \(x\), start by eliminating the constant term that's added or subtracted—18 in this case.
To remove 18, subtract it from both sides of the equation:

• \(3x + 18 - 18 = 48 - 18\)

This simplifies to \(3x = 30\).
The final step is to isolate \(x\) by dividing both sides by the coefficient of \(x\), which is 3:
\(x = \frac{30}{3}\), resulting in \(x = 10\).

Isolation of the variable transforms a complicated equation into a simple equation with the variable alone on one side. This step completes the solution, ensuring you now have the answer to the original equation.