Problem 31 - Algebra for College Students

Question

Solve each equation. \(4(x-3)=-20\)

Step-by-Step Solution

Answer

The solution to the equation is \( x = -2 \).

Step 1: Distribute the 4

To solve the equation, first distribute the 4 across the terms inside the parentheses on the left-hand side. This means multiplying 4 with each term inside: \[ 4(x - 3) = 4x - 12 \]

Step 2: Set the equation equal to -20

Now that the equation is \[ 4x - 12 = -20 \]Keep it in this equation format to prepare for solving for \( x \).

Step 3: Add 12 to both sides

To isolate the term with \( x \), add 12 to both sides of the equation:\[ 4x - 12 + 12 = -20 + 12 \]This simplifies to:\[ 4x = -8 \]

Step 4: Divide both sides by 4

Finally, solve for \( x \) by dividing both sides by 4:\[ \frac{4x}{4} = \frac{-8}{4} \]This results in:\[ x = -2 \]

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions and equations. It involves multiplying a single term by each term within a parenthesis. At the heart of the distributive property is the idea of distribution, where one number is spread across others in an expression.

For instance, in the equation \(4(x-3) = -20\), the distributive property allows us to multiply the 4 with both \(x\) and \(-3\).
This results in the equation: \[ 4x - 12 = -20 \]

• Multiply 4 by \(x\): becomes \(4x\).
• Multiply 4 by \(-3\): becomes \(-12\).

This step is crucial for removing the parentheses, which makes the equation easier to handle. Understanding how to apply the distributive property helps pave the way for further steps in solving linear equations.

Isolating Variables

Isolating variables is an essential step when solving algebraic equations. The goal is to get the variable you are solving for, such as \(x\), on one side of the equation. By isolating \(x\), you make it the subject of the equation which eventually reveals its value.
In our example, once we have expanded the equation to \(4x - 12 = -20\), we begin isolating \(x\) by eliminating other terms around it. We achieve this by adding or subtracting numbers on both sides of the equation.

• Add 12 to both sides: \(4x - 12 + 12 = -20 + 12\).
• Simplify the equation: results in \(4x = -8\).

Through these actions, the term without \(x\) (\(-12\)) is removed, bringing us closer to finding the value of \(x\). Isolating the variable is a methodical process and is key to simplifying complex equations.

Solving for x

Once the variable is isolated, the next step is solving for it, which means finding its numerical value. This involves performing operations to transform the equation into a simple formula that directly states the value of \(x\).
To complete this for \(x\) in our example, you divide both sides of the equation by the coefficient of \(x\). Here, the coefficient is 4 in the equation \(4x = -8\).

• Divide both sides by 4: \(\frac{4x}{4} = \frac{-8}{4}\).
• Simplify: results in \(x = -2\).

This means \(x\) equals \(-2\), concluding the solving process. It's important to ensure each step is correct, as any mistake can lead to incorrect solutions. By carefully following these steps, you successfully solve the equation, confirming that \(x\)'s value makes the entire equation true.