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Chapter 8: Problem 29

In the following exercises, solve each equation. $$ 4(y-2)-3 y=-6 $$

Short Answer

Expert verified
y = 2

Step by step solution

01

- Distribute the 4

Distribute the 4 to both terms inside the parentheses: 4(y-2) becomes 4y - 8, so the equation turns into: 4y - 8 - 3y = -6.
02

- Combine like terms

Combine the like terms on the left side of the equation: 4y - 3y - 8 = -6 leads to y - 8 = -6.
03

- Solve for y

Add 8 to both sides of the equation to isolate y: y - 8 + 8 = -6 + 8 simplifies to y = 2.

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Key Concepts

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

Distributive Property
To start solving the equation, we first apply the distributive property to remove the parentheses. The distributive property states that multiplying a number by a sum or difference is the same as doing each multiplication separately and then adding the results. In this exercise, we distribute the 4 across the terms inside the parentheses: 4(y-2). This becomes: 4y - 8. Now the equation changes to: 4y - 8 - 3y = -6. This step helps simplify the equation, making it easier to work with.
Combining Like Terms
After applying the distributive property, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have 4y and -3y on the left side of the equation. By combining these terms, we get: 4y - 3y = y. So, the equation now looks like: y - 8 = -6. Combining like terms simplifies the equation further, getting us closer to isolating the variable.
Isolating Variables
The final step is to solve for the variable by isolating it. In this equation, we need to isolate y. We do this by performing the same operation on both sides of the equation to keep it balanced. Here, we add 8 to both sides to cancel out the -8 on the left side: y - 8 + 8 = -6 + 8. This simplifies to: y = 2. Now we have successfully solved the equation. Isolating the variable is crucial because it gives us the solution to the equation in the simplest form.

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