/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 191 Solve for the formula for \(y\).... [FREE SOLUTION] | 91影视

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Chapter 2: Problem 191

Solve for the formula for \(y\). Solve the formula \(2 x+3 y=12\) for \(y\).

Short Answer

Expert verified
\(y = 4 - \frac{2x}{3}\)

Step by step solution

01

Start with the given formula

The given formula is: \(2x + 3y = 12\)
02

Isolate the term involving y

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

Solve for y

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

Simplify the fraction

Simplify the fraction by distributing the division across the numerator: \(y = \frac{12}{3} - \frac{2x}{3}\) Thus, \(y = 4 - \frac{2x}{3}\)

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Key Concepts

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

isolating variables
In algebra, isolating the variable means manipulating the equation so that one side of the equation contains the variable we are solving for, and the other side contains only constants and/or other variables. This allows us to solve for the desired variable more easily.
In our given exercise, the target variable to isolate is \( y \).

Here鈥檚 how it's done step-by-step for the equation \( 2x + 3y = 12 \):
  • First, identify the term containing the variable you want to isolate, which is \( 3y \).
  • Subtract \( 2x \) from both sides to isolate \( 3y \) on one side: \( 2x + 3y - 2x = 12 - 2x \), simplifying to \( 3y = 12 - 2x \).
  • This step is crucial as it brings us closer to solving for \( y \).
Practice isolating different variables in various equations to build your confidence and understanding.
rearranging equations
Rearranging equations involves moving terms around to isolate the variable of interest. This often requires applying arithmetic operations such as addition, subtraction, multiplication, and division.

Continuing from where we left off in isolating the variable, we have: \( 3y = 12 - 2x \).

To further rearrange this equation and solve for \( y \), we need to perform the following:
  • Divide both sides of the equation by 3 to neutralize the coefficient of \( y \). This results in: \( y = \frac{12 - 2x}{3} \).
  • At this stage, the equation is already solved for \( y \), but the expression may still be simplified further.
These steps demonstrate how moving terms around and applying arithmetic operations can help isolate the variable and solve the equation.
simplifying fractions
Simplifying fractions is crucial to making expressions easier to understand and work with. In the final step of solving our equation, we simplify the fraction \( \frac{12 - 2x}{3} \):

First, we distribute the division across each term in the numerator:
  • \( y = \frac{12}{3} - \frac{2x}{3} \). This step splits the fraction into more manageable parts.
  • Next, simplify each part of the fraction: \( \frac{12}{3} = 4 \) and \( \frac{2x}{3} \) remains as is.
  • Therefore, we have: \( y = 4 - \frac{2x}{3} \).
This final simplified version of the equation is cleaner and easier to interpret. Whenever working with fractions, always look for opportunities to simplify them to make your equations more straightforward.

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Most popular questions from this chapter

Solve. For each inequality, also graph the solution and write the solution in interval notation. $$ |3 x-4| \geq 2 $$

Solve each inequality, graph the solution, and write the solution in interval notation. \(7 x-8<6\) and \(5 x+7>-3\)

Solve each inequality, graph the solution, and write the solution in interval notation. \(5 x-2 \leq 3 x+4\) and \(3 x-4 \geq 2 x+1\)

Solve each inequality, graph the solution on the number line, and write the solution in interval notation. \(4-3 x \leq-2\) or \(2 x-1 \leq-5\)

Solve. $$ 3|x+2|-5=4 $$
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