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Chapter 2: Problem 258

Solve each linear equation. $$11-4(y-8)=43$$

Short Answer

Expert verified
The solution to the equation is \( y = 0 \).

Step by step solution

01

- Distribute the -4

Expand the equation by distributing the -4 to both terms inside the parentheses: \[ 11 - 4(y - 8) = 43 \] becomes \[ 11 - 4y + 32 = 43 \]
02

- Combine like terms

Now, combine the constants on the left side of the equation: \[ 11 + 32 - 4y = 43 \] which simplifies to \[ 43 - 4y = 43 \]
03

- Isolate the variable term

Subtract 43 from both sides to get the variable term by itself: \[ 43 - 4y - 43 = 43 - 43 \] which becomes \[ -4y = 0 \]
04

- Solve for the variable

Divide both sides by -4 to solve for \( y \): \[ \frac{-4y}{-4} = \frac{0}{-4} \] which simplifies to \[ y = 0 \]

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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 fundamental algebraic principle. It allows you to multiply a term outside a set of parentheses by each term inside. In our exercise, we distribute \( -4 \) across \( y - 8 \). This means \( -4 \) multiplies both \( y \) and \( -8 \). Here's how it looks:

\[ 11 - 4(y - 8) = 43 \]
\[ 11 - 4y + 32 = 43 \]
Now, the equation no longer has parentheses, making it easier to handle.
combining like terms
After distributing, it's time to combine like terms. Like terms are terms that have the same variable part or are constants. In the equation \[ 11 - 4y + 32 = 43 \], the constants are 11 and 32. We add these together:

\[ 11 + 32 = 43 \]
The equation then becomes: \[ 43 - 4y = 43 \]
This step simplifies the equation, reducing the number of terms.
isolating the variable
Now, we want the variable term alone on one side of the equation. We'll subtract 43 from both sides:

\[ 43 - 4y - 43 = 43 - 43 \]
Which simplifies to:
\[ -4y = 0 \]
By isolating the variable term, we set up the equation to solve for the variable in the next step.
basic algebra
In basic algebra, the final step is solving for the variable. We have:

\[ -4y = 0 \]
To solve for \( y \), divide both sides by \( -4 \):

\[ \frac{-4y}{-4} = \frac{0}{-4} \]
Simplifying this:
\[ y = 0 \]
This shows the solution: \( y \) equals zero. Understanding each piece of basic algebra helps in solving more complex equations.

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