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Chapter 2: Problem 34

Solve each formula for the indicated letter. $$ 3 x-y=7, \text { for } y $$

Short Answer

Expert verified
y = 3x - 7.

Step by step solution

01

- Identify the equation

The given equation is: 3x - y = 7. We need to solve for y.
02

- Isolate the term involving y

To isolate the term involving y, move the term with x to the other side by subtracting 3x from both sides: 3x - y - 3x = 7 - 3x. This simplifies to: -y = 7 - 3x.
03

- Solve for y

To solve for y, multiply both sides of the equation by -1: -1(-y) = -1(7 - 3x). This simplifies to: y = 3x - 7.

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Key Concepts

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

Isolating Variables
When solving equations, one of the vital steps is isolating the variable you need to solve for. This means rearranging the equation so that the variable is by itself on one side of the equation. In our example, we needed to solve for the variable y in the equation \(3x - y = 7\).

To isolate y, we started by moving the term involving x to the other side. This was achieved by subtracting \(3x\) from both sides of the equation. This step ensures that the term involving y remains on one side by itself.

In mathematical terms, isolating a variable often involves:
  • Addition or subtraction
  • Multiplication or division
  • Using inverse operations
By systematically applying these operations, you can isolate any variable in an equation.
Linear Equations
Linear equations are equations of the first degree, meaning they involve variables raised only to the power of one. They have the general form:\[Ax + By = C\]

In our exercise, \(3x - y = 7\) is a linear equation. Here鈥檚 why:
  • The variables (x and y) are to the first power.
  • The equation forms a straight line when graphed.

Understanding the properties of linear equations is crucial for solving them effectively. They are easier to work with than higher-degree equations, and their solutions involve straightforward algebraic manipulation.
Algebraic Manipulation
Algebraic manipulation involves using mathematical operations to rearrange and simplify equations. This skill is essential for solving for a specific variable, as demonstrated in our solution. We used several algebraic manipulation steps to isolate and solve for y:
  • Identifying variables and constants in the equation.
  • Using inverse operations to isolate the variable.
  • Simplifying the remaining terms.

For instance, we first identified that -y was not isolated and subtracted \(3x\) from both sides: \(-y = 7 - 3x\).

Next, to solve for y, we multiplied both sides by -1 because the original equation had -y. This means:\(-1(-y) = -1(7 - 3x)\), which simplifies to \(y = 3x - 7\).

Mastering algebraic manipulation strengthens your ability to handle various equations in algebra and beyond.

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Most popular questions from this chapter

In exchange for opening a new credit account, Macy's Department Stores 'subtracts \(10 \%\) from all purchases made on the day the account is established. Julio is opening an account and has a coupon for which he receives \(10 \%\) off the first day's reduced price of a camera. If Julio's final price is \(\$ 77.75,\) what was the price of the camera before the two discounts?

The perimeter of a rectangle is \(101.74 \mathrm{cm} .\) If the length is \(4.25 \mathrm{cm}\) longer than the width, find the dimensions of the rectangle.

Translate to an inequality. Normandale Community College is no more than 15 mi away.

After 9 quizzes, Blythe's average is 84. Is it possible for Blythe to improve her average by two points with the next quiz? Why or why not?

Use an inequality and the five-step process to solve each problem. RJ's Plumbing and Heating charges \(\$ 55\) plus \(\$ 40\) per hour for emergency service. Gary remembers being billed over \(\$ 100\) for an emergency call. How long was RJ's there?
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