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Chapter 2: Problem 120

Solve each equation requiring simplification. $$-12(d-5)-29=43$$

Short Answer

Expert verified
d = -1

Step by step solution

01

Distribute

Distribute the \(-12\) inside the parentheses: \(-12(d-5)-29=43\) becomes \(-12d + 60 - 29 = 43\)
02

Combine Like Terms

Combine the constants on the left side: \(60 - 29\) results in \(-12d + 31 = 43\)
03

Isolate the Variable Term

Subtract \(31\) from both sides to isolate the term with the variable: \(-12d + 31 - 31 = 43 - 31\) simplifies to \(-12d = 12\)
04

Solve for the Variable

Divide both sides by \(-12\) to solve for \(d\): \(-12d \/ -12 = 12 \/ -12\) results in \(d = -1\)

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Key Concepts

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

Distributive Property
The distributive property is a useful tool when solving linear equations. It allows you to multiply a single term across terms inside parentheses. For the equation \(-12(d-5) - 29 = 43\), we start by distributing \(-12\) across \(d-5\). This means multiplying -12 by both \(d\) and \(-5\). Doing the math, we get: \(-12 \times d = -12d\) and \(-12 \times (-5) = 60\). Combining these, the equation becomes \(-12d + 60 - 29 = 43\).

Remember, distributing helps eliminate parentheses, making the equation simpler to manage. It's important to apply it correctly to avoid errors in solving the equation.
Combining Like Terms
After distributing, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power, or constants without variables. In our equation \(-12d + 60 - 29 = 43\), the constants are \(60\) and \(-29\). By combining these, we get: \(60 - 29 = 31\).

The new equation then is: \(-12d + 31 = 43\). Combining like terms simplifies the equation, reducing the number of terms and making it easier to isolate the variable.
Isolating Variables
To solve the equation for the variable, we need to isolate the variable term. Starting with \(-12d + 31 = 43\), we first get rid of the constant \(31\) on the left side. This is done by subtracting \(31\) from both sides: \(-12d + 31 - 31 = 43 - 31\), leaving us with \(-12d = 12\).

The final step is to isolate \(d\) by dividing both sides by \(-12\): \(-12d / -12 = 12 / -12\), which simplifies to \(d = -1\).

Isolating variables helps us find the value of the variable, providing the solution to our equation.

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