/*! 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 109 Solve each equation, if possible... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 109

Solve each equation, if possible. $$ -\frac{2}{3} z+4=8 $$

Short Answer

Expert verified
The solution is \(z = -6\).

Step by step solution

01

Move Constant to the Other Side

Start by isolating the term with the variable on one side of the equation. Subtract 4 from both sides: \[-\frac{2}{3}z + 4 - 4 = 8 - 4\]Simplifying this gives us:\[-\frac{2}{3}z = 4\]
02

Eliminate the Fraction Coefficient

We need to get rid of the fractional coefficient \(-\frac{2}{3}\). Multiply both sides of the equation by \(-\frac{3}{2}\), the reciprocal of \(-\frac{2}{3}\): \[(-\frac{3}{2})(-\frac{2}{3})z = 4 \times (-\frac{3}{2})\]This simplifies to:\[z = -6\]

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.

Fractions in Equations
Solving equations with fractions can be daunting, but understanding how fractions work can make them less intimidating. A fraction consists of a numerator and a denominator. In an equation, fractions often serve as coefficients for variables. In this particular problem, we have \[ -\frac{2}{3} z + 4 = 8 \]where \(-\frac{2}{3}\) is the coefficient of \(z\). To effectively handle equations with fractions, it's crucial to standardize them first by isolating the fraction on one side if needed.
  • Start by focusing on the term involving the fraction.
  • Shift any constants away so you can work with the fraction independently.
Understanding how to manipulate fractions can make these types of problems far more manageable.
Isolation of Variables
The principle of isolating the variable is one of the foundational strategies in solving linear equations. Essentially, you aim to get the variable by itself on one side of the equation. This allows you to see clearly what the solution is. Using our example:\[-\frac{2}{3}z + 4 = 8 \]
  • We begin by moving the constant 4 away from the variable term. This is done through subtraction from both sides resulting in:
  • \(-\frac{2}{3}z = 4\).
Achieving isolation supports a more straightforward application of subsequent operations to solve for the variable. It's much like peeling layers off an onion. Keep performing operations that simplify the equation until the variable stands alone.
Reciprocal Multiplication
Reciprocal multiplication is a clever way to handle fractional coefficients. The reciprocal of a fraction is simply flipping its numerator and denominator. In the given equation \[-\frac{2}{3}z = 4\], the reciprocal of \(-\frac{2}{3}\) is \(-\frac{3}{2}\). By multiplying both sides of the equation with this reciprocal:
  • The fractional coefficient is neutralized leading to \(z\) alone on one side:
  • \((-\frac{3}{2})(-\frac{2}{3})z = 4 \times (-\frac{3}{2})\)
  • Simplifies to \(z = -6\).
Use reciprocal multiplication as a tool to make fractions disappear, leaving you with a more straightforward equation. Once you master this technique, dealing with fractions will become second nature.

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

How much water should be added to 20 ounces of a \(15 \%\) solution of alcohol to dilute it to a \(10 \%\) alcohol solution?

Solve each inequality. Write the solution set in interval notation and graph it. $$ \frac{3 a-4}{-5}<\frac{3 a+15}{-5} $$

Solve each inequality or compound inequality. Write the solution set in interval notation and graph it. $$ \frac{a}{-25}+3<3 $$

What percent of 200 is \(30 ?\)

Dry mixture problems. Epiphanny Prince, of New York, scored 113 points in a high school game on February \(1,2006,\) breaking a national prep record that was held by Cheryl Miller. Prince made 46 more 2 -point baskets than 3 -point baskets, and only 1 free throw. How many 2 -point and 3 -point baskets did she make?
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Applied 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.