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Chapter 2: Problem 77

Solve each equation, and check the solution. $$ 4(3 x-1)+4=-36 $$

Short Answer

Expert verified
x = -3

Step by step solution

01

Distribute the 4

First, distribute the 4 across the terms inside the parentheses: \[ 4(3x - 1) + 4 \rightarrow 4 \times 3x + 4 \times (-1) + 4 \] This simplifies to: \[ 12x - 4 + 4 \]
02

Simplify the Expression

Next, combine the like terms on the left side: \[ 12x - 4 + 4 = 12x \] So the equation now is: \[ 12x = -36 \]
03

Solve for x

To isolate x, divide both sides of the equation by 12: \[ \frac{12x}{12} = \frac{-36}{12} \] This simplifies to: \[ x = -3 \]
04

Check the Solution

Substitute the value of x back into the original equation to verify: \[ 4(3(-3) - 1) + 4 = -36 \] Simplify inside the parentheses: \[ 4(-9 - 1) + 4 = -36 \] Combine the terms: \[ 4(-10) + 4 = -36 \] Perform multiplication and addition: \[ -40 + 4 = -36 \] \[ -36 = -36 \] Since both sides of the equation are equal, the solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Using the Distributive Property
To solve linear equations, understanding the distributive property is crucial. The distributive property allows you to multiply a single term by each term within parentheses. For example, in the equation \( 4(3x - 1) + 4 = -36 \), you need to distribute the 4 across each term inside the parentheses: \[ 4 \times 3x + 4 \times (-1) + 4 \] becomes \[ 12x - 4 + 4 \]. Notice how each term inside the parentheses is multiplied by 4. This simplification step is essential because it allows us to remove the parentheses and work with simpler algebraic expressions.
Combining Like Terms
Once you have used the distributive property, 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 our example, after distributing, we have the expression \[ 12x - 4 + 4 \]. Here, \(-4\) and \(+4\) are like terms since they are both constants (numbers without variables). Combining these like terms, we get \[ 12x \]. Combining like terms helps to further simplify the equation and make it easier to solve. Always remember to look for terms that can be combined to simplify your equation.
Isolating the Variable
The ultimate goal when solving linear equations is to isolate the variable on one side of the equation. In this example, we have simplified the equation to \[ 12x = -36 \]. To isolate \( x \), you need to perform operations that will leave \( x \) alone on one side of the equation. In this case, you can isolate \( x \) by dividing both sides of the equation by 12: \[ \frac{12x}{12} = \frac{-36}{12} \]. This simplifies to \( x = -3 \). By isolating the variable, you find the solution to the equation. Always ensure to check your solution by substituting the value back into the original equation to verify it.

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