/*! 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 43 Write each equation in the form ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 43

Write each equation in the form \(y=m x+b.\) $$-2 y=14 x+\frac{1}{2}(6 x+12)$$

Short Answer

Expert verified
\(y = \frac{17}{2} x + 3\)

Step by step solution

01

- Distribute the Fraction

Distribute \ \(\frac{1}{2}\) to both terms inside the parentheses: \ \(\frac{1}{2}(6 x + 12) = \frac{1}{2} \cdot 6 x + \frac{1}{2} \cdot 12\). This simplifies to \ \(3 x + 6\).
02

- Combine Like Terms

Combine the like terms from the equation \ \(-2 y = 14 x + 3 x + 6\). This simplifies to \ \(-2 y = 17 x + 6\).
03

- Isolate the y term

Divide each term by \(-2\) to solve for \(y\). \ \(y = \frac{-17 x - 6}{-2} = \frac{-17 x}{-2} + \frac{-6}{-2} = \frac{17 x}{2} + 3\).
04

- Final Equation in Slope-Intercept Form

Simplify to get the equation in the form \(y = mx + b\). \ \(y = \frac{17}{2} x + 3\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Linear Equations
Linear equations are equations of the first order. They make a straight line when graphed, which means they have no curves, bends, or intersections with themselves. The most common form of a linear equation is the slope-intercept form, written as:
  • \(y = mx + b\)
Here, m represents the slope of the line, and b represents the y-intercept, where the line crosses the y-axis.
Understanding how to manipulate and solve these equations is key in algebra.
For the given exercise, converting an equation like \(-2y = 14x + \frac{1}{2}(6x + 12)\) into the slope-intercept form involves a series of crucial steps that include distribution, combination of terms, and isolation of the desired variable.
Mastering Algebraic Manipulation
Algebraic manipulation involves rearranging equations to simplify and solve them. Let's go over the steps from the exercise to see how this works in practice.
  • Step 1: Distribute the fraction.
    The term \(\frac{1}{2}(6x + 12)\) means we distribute \(\frac{1}{2}\) over both terms inside the parentheses:
    \(\frac{1}{2} \times 6x + \frac{1}{2} \times 12 = 3x + 6\).

    • This is a crucial step to convert complex equations into simpler forms.
  • Step 2: Combine like terms.
    Here, we combine \(14x\) and \(3x\), resulting in:
    \(-2y = 17x + 6\).

  • Step 3: Isolate the desired term, y.
    We divide every term by \(-2\), while keeping algebraic integrity:
    \(y = \frac{-17x - 6}{-2} = \frac{17x}{2} + 3\).

Mastering these steps allows you to handle more complex algebraic challenges with confidence.
Solving Equations Effectively
Solving equations is the process of finding the value of the unknown that makes the equation true. Let鈥檚 look at how we solved the original problem.
Given the final form: \(y = \frac{17}{2}x + 3\), this result was achieved through precise algebraic manipulation.
  • First, distribute fractions to simplify expressions.
  • Next, combine like terms for clarity.
  • Finally, isolate the variable of interest, y, by dividing each term by a common factor.
Applying these principles ensures logical steps are followed in algebra, resulting in correct solutions.
In summary, solving equations effectively requires understanding linear equations, mastering algebraic manipulation, and following logical steps to isolate and solve for unknowns.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Lindsey said, "T'm thinking of a number. If I multiply it by \(5,\) subtract \(4,\) and then multiply the result by \(2,\) I get \(62 .\) What is my number?

Graph each inequality. $$y<3 x+7$$

Megan is writing a computer game in which a player stands on the balcony of a haunted house and drops water balloons on ghosts below. The player chooses where the balloon will land and then launches it. Since water splatters, Megan鈥檚 game gives the player points if a ghost is anywhere within a square centered where the balloon lands. The square extends 15 units beyond the center in all four directions. That is, if both of the ghost鈥檚 coordinates are 15 units or less from the center, the player has scored a hit. Suppose the balloon lands at \((372,425) .\) The nearest ghost has the coordinates \((x, y),\) and it counts as a hit by the game. Use inequalities to describe the possible values for \(x\) and \(y .\) (Hint: You will need two inequalities, one for \(x\) and one for \(y .\) Should you say "and" or "or' between them?)

Consider the inequality \(x^{3} \leq 27.\) a. Express the solution of \(x^{3} \leq 27\) as an inequality. b. Graph the solution on a number line.

Rewrite each expression as simply as you can. $$m^{7} \cdot m^{-5} \cdot b^{5}$$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.