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Chapter 8: Problem 147

Solve the equations for the variable. $$3 y-4=12-y$$

Short Answer

Expert verified
y = 4

Step by step solution

01

- Move the variable terms to one side

First, add y to both sides of the equation to move all the variable terms to one side: \[ 3y - 4 + y = 12 - y + y \] which simplifies to \[ 4y - 4 = 12 \].
02

- Isolate the variable term

Next, add 4 to both sides of the equation to isolate the term with the variable: \[ 4y - 4 + 4 = 12 + 4 \] which simplifies to \[ 4y = 16 \].
03

- Solve for the variable

To find the value of y, divide both sides of the equation by 4: \[ \frac{4y}{4} = \frac{16}{4} \] which simplifies to \[ y = 4 \].

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Key Concepts

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

Equation Simplification
In any equation, simplifying each side is often the first step. Simplification makes complex equations easier to solve. It means combining like terms, which are terms that have the same variable raised to the same power.

Consider the equation from our exercise: 3 y - 4 = 12 - y. To simplify, we need to get all variable terms (terms with 'y') on one side and constants on the other. We started with 3y - 4 + y = 12 - y + y. The + y and - y cancel each other out, leaving: 4 y - 4 = 12. This new equation is simpler and makes the next steps more straightforward.
Isolating Variables
Once an equation is simplified, our goal is to isolate the variable we are solving for, usually by performing the same operation on both sides of the equation.

Let's revisit our simplified equation: 4y - 4 = 12. Our next step is to get y by itself on one side of the equation. We do this by undoing the operations around y. First, we add 4 to both sides to remove the -4: 4y - 4 + 4 = 12 + 4.

This isolates the term with the variable y, giving us 4y = 16. The equation is now even simpler, and we can easily solve for y.
Prealgebra Techniques
Prealgebra involves basic mathematical operations such as addition, subtraction, multiplication, and division. These are foundational techniques for solving equations. In this exercise, we used several prealgebra techniques:
  • Combining like terms: Adding y to both sides to combine variable terms into a single term.
  • Adding or subtracting constants: Adding 4 to both sides to eliminate the constant term next to y.
  • Using basic division: Dividing both sides by 4 to solve for y.

In the final step of our example: 4y = 16, dividing both sides by 4 results in y = 4.

By mastering these basic techniques, you'll be able to efficiently tackle more complex mathematical challenges.

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Most popular questions from this chapter

In the following exercises, translate into an equation and solve. Pilar drove from home to school and then to her aunt鈥檚 house, a total of 18 miles. The distance from Pilar鈥檚 house to school is 7 miles. What is the distance from school to her aunt鈥檚 house?

In the following exercises, solve each equation by clearing the decimals. $$ 0.6 p-1.9=0.78 p+1.7 $$

In the following exercises, trach English sentence into an algebraic equation and then solve it. Four less than \(n\) is 13

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