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Chapter 2: Problem 36

Solve each formula for the indicated letter. $$ 3 x+2 y=12, \text { for } y $$

Short Answer

Expert verified
y = 6 - \frac{3x}{2}

Step by step solution

01

Rewrite the equation

Start with the original equation: \[3x + 2y = 12\]
02

Isolate the term containing y

Subtract \(3x\) from both sides to isolate the term with \(y\): \[2y = 12 - 3x\]
03

Solve for y

Divide both sides by 2 to solve for \(y\): \[y = \frac{12 - 3x}{2}\]
04

Simplify (if necessary)

Simplify the fraction (if possible): \[y = 6 - \frac{3x}{2}\]

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Key Concepts

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

linear equations
A linear equation is an equation where the highest power of the variable(s) is one. This means the graph of a linear equation is a straight line. The general form of a linear equation in two variables, x and y, is: \[ax + by = c\]where \(a\), \(b\), and \(c\) are constants. In the given exercise, we have the linear equation: \[3x + 2y = 12\]Here, the coefficients are 3 and 2 for x and y respectively, and the constant term is 12. Linear equations are foundational in algebra because they are straightforward and lay the groundwork for understanding more complex equations.
isolate variable
Isolating a variable means we want to get one of the variables alone on one side of the equation. It's a common step when solving equations, especially when solving for a specific variable. In the provided exercise, we are solving for y. Here's how we isolate y:- First, start with the original equation: \[3x + 2y = 12\]- Next, subtract 3x from both sides to isolate the term with y: \[2y = 12 - 3x\]This step is crucial because it helps us get closer to having y by itself.
simplifying fractions
Simplifying fractions makes them easier to work with and understand. In our exercise, we need to simplify the fraction to solve for y. Let's recall where we left off:\[2y = 12 - 3x\]To solve for y, divide both sides by 2:\[y = \frac{12 - 3x}{2}\]Now, break it down step by step. The expression \(\frac{12 - 3x}{2}\) can be separated as:\[y = \frac{12}{2} - \frac{3x}{2}\]This simplifies to:\[y = 6 - \frac{3x}{2}\]By simplifying fractions, we make the solution clearer and more concise.

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Most popular questions from this chapter

Use an inequality and the five-step process to solve each problem. Following the guidelines of the Food and Drug Administration, Dale tries to eat at least 5 servings of fruits or vegetables each day. For the first \(\operatorname{six}\) days of one week, she had \(4,6,7,4,6,\) and 4 servings. How many servings of fruits or vegetables should Dale eat on Saturday, in order to average at least 5 servings per day for the week?

Use an inequality and the five-step process to solve each problem. Rhetoric Advertising is a directmail company. It determines that for a particular campaign, it can use any envelope with a fixed width of \(3 \frac{1}{2}\) in. and an area of at least \(17 \frac{1}{2}\) in \(^{2} .\) Determine (in terms of an inequality) those lengths that will satisfy the company constraints.

Use an inequality and the five-step process to solve each problem. The perimeter of a rectangular swimming pool is not to exceed 70 ft. The length is to be twice the width. What widths will meet these conditions?

six people. One-third, one-fourth, one-eighth, and one-fifth of the apples are given to four people, respectively. The fifth person gets ten apples, and one apple remains for the sixth person. Find the original number of apples in the basket.

Solve using the addition and multiplication principles. $$ \frac{2}{3}\left(\frac{7}{8}-4 x\right)-\frac{5}{8}<\frac{3}{8} $$
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