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Chapter 1: Problem 64

Solve. $$ 3(y+5)=8 y $$

Short Answer

Expert verified
y = 3

Step by step solution

01

- Distribute the 3 on the left side

Apply the distributive property to the left-hand side of the equation: 3(y + 5) = 3y + 15
02

- Write the new equation

Now the equation is: 3y + 15 = 8y
03

- Move terms involving y to one side

Subtract 3y from both sides of the equation to get all terms involving y on one side: 15 = 5y
04

- Solve for y

Divide both sides by 5 to isolate y: y = 3

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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 concept in algebra. It allows you to multiply a single term by a sum inside parentheses. In this exercise, we start with the equation \[3(y + 5) = 8y\]
With the distributive property, we will distribute the 3 to both terms inside the parentheses: \[3(y + 5) = 3 \times y + 3 \times 5 = 3y + 15\]
This step is crucial because it simplifies the equation and makes it easier to solve. Remember, anytime you see a number outside parentheses, you'll use the distributive property to spread it across the terms inside.
Isolating Variables
After using the distributive property, the equation becomes \[3y + 15 = 8y\]
Our goal is to solve for the variable y by isolating it on one side of the equation. We'll do this by moving all terms involving y to the same side.
We start by subtracting 3y from both sides:
\[3y + 15 - 3y = 8y - 3y\]
This simplifies to: \[15 = 5y\]
Now, y is nearly isolated, making it easier to solve for y in the next step.
Basic Algebra
In basic algebra, solving equations often involves simple operations like addition, subtraction, multiplication, and division. After isolating the variable terms, we continue from where we left off: \[15 = 5y\]
To find y, divide both sides of the equation by 5: \[\frac{15}{5} = \frac{5y}{5}\]
Simplifying this, we get: \[y = 3\]
Always remember to perform the same operation on both sides of the equation to maintain the balance. Basic algebra techniques are about manipulating equations by applying these operations correctly. Understanding these steps helps solve more complex equations in the future.

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