/*! 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 85 Solve each equation for \(y\). ... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 85

Solve each equation for \(y\). $$ 3 x+4 y=12 $$

Short Answer

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

Step by step solution

01

- Isolate the term with y

Start by isolating the term with y. Subtract 3x from both sides of the equation: \[ 3x + 4y - 3x = 12 - 3x \] which simplifies to \[ 4y = 12 - 3x \]
02

- Solve for y

Divide both sides of the equation by 4 to solve for y: \[ y = \frac{12 - 3x}{4} \] Simplify to get: \[ y = 3 - \frac{3x}{4} \]

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Key Concepts

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

Isolating Variables
To solve linear equations like the one in our exercise, we start by isolating the variable we want to solve for. In this case, we isolate the variable y. Isolating a variable means getting it alone on one side of the equation. This helps us understand its value in relation to the other terms. Here, we begin by isolating the term with y. We do this by subtracting 3x from both sides of the equation:
ext: \[ 3x + 4y = 12 \] becomes \[ 4y = 12 - 3x \] By subtracting 3x, we moved it to the other side of the equation, making it easier to focus on y.
Algebraic Manipulation
Algebraic manipulation involves using various techniques to simplify equations and expressions so we can solve for variables. It includes addition, subtraction, multiplication, and division to keep the equation balanced. In our problem, we used the subtraction of 3x (algebraic manipulation) to help isolate y.
  • First, we subtracted 3x from both sides: \[ 3x + 4y - 3x = 12 - 3x \]
  • Next, we simplified the equation to: \[ 4y = 12 - 3x \]
  • Then, we divided both sides by 4 to solve directly for y: \[ y = \frac{12 - 3x}{4} \]
Algebraic manipulation is crucial because it helps transform equations into more straightforward, solvable forms.
Linear Equations
Linear equations are equations where the highest power of the variable is 1. They have a straight-line graph and take the form Ax + By = C. In our exercise, the equation is 3x + 4y = 12, which is a typical linear equation. Solving these equations involves finding the values of x and y that make the equation true. We simplified our linear equation step-by-step:
  • Starting with \[ 3x + 4y = 12 \]
  • We isolated y to get: \[ 4y = 12 - 3x \]
  • Then, finally got the solution: \[ y = 3 - \frac{3x}{4} \]
Understanding linear equations and their solutions is fundamental in algebra because they represent many real-world scenarios.

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Most popular questions from this chapter

Find the slope of the line through each pair of points.\(\left(\text {Hint:} \frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b} \div \frac{c}{d}\right)\). $$ \left(-\frac{2}{9}, \frac{5}{18}\right) \text { and }\left(\frac{1}{18},-\frac{5}{9}\right) $$

For each situation, (a) write an equation in the form \(y=m x+b,(b)\) find and interpret the ordered pair associated with the equation for \(x=5,\) and \((c)\) answer the question. A ticket for the 2010 Troubadour Reunion, featuring James Taylor and Carole King. costs \(\$ 112.50 .\) A parking pass costs \(\$ 12 .\) (Source: Ticketmaster.) Let \(x\) represent the number of tickets and \(y\) represent the cost. How much does it cost for 2 tickets and a parking pass?

Which of the following forms of the slope formula are correct? Explain. A. \(m=\frac{y_{1}-y_{2}}{x_{2}-x_{1}}\) B. \(m=\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\) C. \(m=\frac{x_{2}-x_{1}}{y_{2}-y_{1}}\) D. \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

For each function, find \((a) f(2)\) and \((b) f(-1) .\) See Examples 4 and \(5 .\) $$ f=\\{(-1,-5),(0,5),(2,-5)\\} $$

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ 4 x-3 y=8 $$
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