Problem 40 - Algebra 1

Question

Solve the equation. $$9 x-5(3 x-12)=30$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Expand the Parenthesis

Multiply the numbers in the parenthesis by the number outside, which leads to the equation: \(9x - 15x + 60 = 30 \)

Step 2: Simplify the Equation

Combine like terms on the left side of the equation: \(-6x + 60 = 30 \)

Step 3: Isolate x

Subtract 60 from both sides of the equation: \(-6x = -30 \)

Step 4: Solve for x

Divide both sides of the equation by -6 to get \(x = 5\)

Key Concepts

Distributive Property

The distributive property is a powerful tool when solving linear equations that involve parentheses. It allows you to distribute a single term across terms inside the parentheses, simplifying equations for easier manipulation.

In the original equation, \(9x - 5(3x - 12) = 30\), the number \(-5\) needs to be distributed across the terms inside the parentheses, \(3x - 12\).

Apply the distributive property by multiplying \(-5\) with each term inside:

• \(-5 \times 3x = -15x\)
• \(-5 \times (-12) = +60\) (note the double negative becomes a positive)

Thus, the expanded form becomes \(9x - 15x + 60 = 30\).

This step is crucial because it sets the stage for combining like terms, reflecting one of the core uses of the distributive property.

Combining Like Terms

Once distributive property is applied, the next step is to simplify the equation by combining like terms. Like terms are terms that have the same variable raised to the same power.

In the equation \(9x - 15x + 60 = 30\), you can see two terms with the variable \(x\): \(9x\) and \(-15x\).

Combine these by adding or subtracting their coefficients:

• \(9x - 15x = -6x\)

Now, the equation simplifies to \(-6x + 60 = 30\).

By simplifying like this, you reduce complexity and bring the equation closer to a form where you can solve for the variable easily.

Isolating Variables

Isolating the variable is often the final move in solving an equation. The goal is to get the variable by itself on one side of the equation, usually achieved through inverse operations.

When we have \(-6x + 60 = 30\), start by eliminating the constant term on the side with the variable. Do this by subtracting \(60\) from both sides:

• \(-6x + 60 - 60 = 30 - 60\)
• This simplifies to \(-6x = -30\)

Next, isolate \(x\) by dividing each side by \(-6\), the coefficient of \(x\):

• \(x = \frac{-30}{-6}\)

This results in \(x = 5\).

Isolating the variable ensures you've solved for the correct value, making it possible to verify your solution.