Problem 36 - Algebra 1

Question

Solve the equation $$ 8=-\frac{2}{3}(2 x-6) $$

Step-by-Step Solution

Answer

The solution of the equation \( 8=-\frac{2}{3}(2 x-6) \) is \( x = -3 \)

Step 1: Distributive Property

Apply the distributive property of multiplication over subtraction to solve inside the parentheses. Therefore, the equation becomes \( 8=-\frac{2}{3}*2x + \frac{2}{3}*6 \). This simplifies to \( 8=-\frac{4}{3}x + 4 \).

Step 2: Isolate the x variable

Start to isolate x by subtracting 4 from both sides of the equation. Therefore, the equation becomes \( 8-4=-\frac{4}{3}x + 4-4 \) which simplifies to \( 4 = -\frac{4}{3}x \).

Step 3: Solve for x

Now, solve for x by dividing each side by -4/3 (or multiplying by -3/4). The equation becomes \( 4*-\frac{3}{4} = x \) simplifying further to \( x=-3 \).

Step 4: Checking the solution

Substitute x = -3 into the original equation to verify that it is correct solution. Left-hand side \( = 8 \) and right-hand side \( = -\frac{2}{3}(2*(-3)-6) = -\frac{2}{3}*(-6) = 8 \). Since left-hand side equals right-hand side, x = -3 is the solution.

Key Concepts

Distributive Property

The distributive property is a fundamental mathematical principle. It helps when you need to multiply a single term across terms inside parentheses. Apply it by multiplying the outer term with each inner term separately.
In our equation, we have to handle \(-\frac{2}{3}(2x - 6)\). Distributing \(-\frac{2}{3}\) to both \(2x\) and \(-6\) gives:

• \(-\frac{2}{3} \times 2x = -\frac{4}{3}x\)
• \(-\frac{2}{3} \times (-6) = 4\) (a negative times a negative is a positive)

Now, substitute these results back to get:\(8 = -\frac{4}{3}x + 4\)

Isolate Variable

Isolating the variable is crucial to solving equations. It involves getting the variable (in this case, \(x\)) on one side of the equal sign.
Begin with: \(8 = -\frac{4}{3}x + 4\). First, remove the constant on the right side by subtracting 4 from both sides. This simplifies to:

• \(8 - 4 = -\frac{4}{3}x + 4 - 4\)
• \(4 = -\frac{4}{3}x\)

This step ensures the equation focuses solely on the variable with a numerical coefficient, making the manipulation simpler.

Checking Solutions

Checking your solutions is vital to ensure your calculations are correct. This process involves substituting your solution back into the original equation to verify.
For this exercise, substitute \(x = -3\) back into \(8 = -\frac{2}{3}(2x - 6)\):

• First, calculate inside the parentheses: \(2(-3) - 6 = -6 - 6 = -12\)
• Multiply: \(-\frac{2}{3} \times (-12) = 8\)

The left-hand side also equals 8, confirming \(x = -3\) is indeed the correct solution.

Division by Fractions

Dividing by fractions can be tricky but becomes easier with practice. You can divide by a fraction by multiplying by its reciprocal. For instance, with \(4 = -\frac{4}{3}x\), you want to isolate \(x\).
Divide both sides by the coefficient of \(x\), which is \(-\frac{4}{3}\):

• The reciprocal of \(-\frac{4}{3}\) is \(-\frac{3}{4}\)
• Multiply both sides by \(-\frac{3}{4}\) to solve for \(x\)
• \(4 \times -\frac{3}{4} = x\)
• \(x = -3\)

This conversion simplifies the calculations, achieving isolation of the variable.