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Chapter 2: Problem 268

Solve each linear equation. $$12+2(5-3 y)=-9(y-1)-2$$

Short Answer

Expert verified
The solution is \( y = -5 \).

Step by step solution

01

Distribute on both sides

Apply the distributive property to both sides of the equation. Expand both expressions by multiplying the factors:\[ 12 + 2(5 - 3y) = -9(y - 1) - 2 \] becomes \[ 12 + 10 - 6y = -9y + 9 - 2 \]
02

Combine like terms

Combine the constant terms on both sides of the equation:\[ 22 - 6y = -9y + 7 \]
03

Collect variable terms on one side

Add 9y to both sides to collect all variable terms on one side:\[ 22 - 6y + 9y = 7 \] which simplifies to \[ 22 + 3y = 7 \]
04

Isolate the variable term

Subtract 22 from both sides to isolate the term with the variable:\[ 3y = 7 - 22 \] which simplifies to \[ 3y = -15 \]
05

Solve for the variable

Divide both sides by 3 to solve for the variable:\[ y = \frac{-15}{3} \] which simplifies to \[ y = -5 \]

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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 allows you to distribute a single term across terms inside parentheses. It is essential for simplifying expressions. In our example, we start with the equation
$$ 12 + 2(5 - 3y) = -9(y - 1) - 2 $$
We apply the distributive property:
  • Multiply 2 by each term inside the parentheses: $$ 2 \times 5 $$ and $$ 2 \times -3y $$
  • Multiply -9 by each term inside the parentheses: $$ -9 \times y $$ and $$-9 \times -1$$
Transforming the equation into: $$ 12 + 10 - 6y = -9y + 9 - 2 $$
Combining Like Terms
Combining like terms involves adding or subtracting terms that have the same variables. This step helps simplify the equation. After distributing, the equation looks like:
$$ 12 + 10 - 6y = -9y + 9 - 2 $$
We then combine the constants and the variable terms separately:
  • Add constants $$12 + 10$$ and combine constant terms on the right $$9 - 2$$
  • The equation becomes: $$ 22 - 6y = -9y + 7 $$
Now, both sides of the equation are simplified, making it easier to isolate the variable.
Isolating Variables
Isolating the variable means moving all the variable terms to one side of the equation. Here, we aim to get all terms with y on one side. We start from:
$$ 22 - 6y = -9y + 7 $$
Add 9y to both sides to combine all variable terms:
  • $$ 22 - 6y + 9y = 7 $$
  • Simplifies to: $$ 22 + 3y = 7 $$
The variable is now isolated on one side, making the next steps straightforward.
Solving for Variables
In this final step, we solve for the variable by isolating it completely. Our goal is to have y alone on one side:
Start with:
$$ 22 + 3y = 7 $$
Subtract 22 from both sides:
  • $$ 3y = 7 - 22 $$
  • Simplifies to: $$ 3y = -15 $$
Divide both sides by 3: $$ y = \frac{-15}{3} $$
which simplifies to: $$ y = -5 $$
By following these steps, you've successfully solved for the variable y.

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