/*! 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 70 Solve each equation. $$ 3 k+... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 70

Solve each equation. $$ 3 k+8=5 $$

Short Answer

Expert verified
The solution is \(k = -1\).

Step by step solution

01

Isolate the Variable Term

Start by getting the term with the variable on one side by itself. Subtract 8 from both sides of the equation to move the constant term to the right. \[3k + 8 - 8 = 5 - 8\]This simplifies to:\[3k = -3\]
02

Solve for the Variable

Now that we have isolated the term with the variable, solve for \(k\) by dividing both sides by 3 to find the value of \(k\):\[\frac{3k}{3} = \frac{-3}{3}\]Which simplifies to:\[k = -1\]

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.

Isolate the Variable
In the process of solving linear equations, it is crucial to "isolate the variable". This means getting the variable term on one side of the equation all by itself.
To do this, we must first eliminate any numbers or terms that are added or subtracted to the variable term.
In our example equation, \(3k + 8 = 5\), we begin by removing the constant term next to the variable, which is the +8.

  • Subtract 8 from both sides of the equation: \(3k + 8 - 8 = 5 - 8\).
  • This results in \(3k = -3\).
This step is about making the surrounding equation simpler, allowing a clear path to find the value of the variable. Once isolated, the variable term can be easily manipulated to solve the equation.
Inverse Operations
"Inverse operations" are fundamental tools used to solve equations. They are operations that undo each other, such as addition vs. subtraction or multiplication vs. division.
These operations help us move terms around and simplify equations by doing the opposite operation to both sides of the equation.

  • If you add something to one side, subtract the same thing from both sides to maintain balance.
  • If you multiply, divide both sides by the same value to undo the multiplication.
In the example equation, after isolating the variable term, we addressed the multiplication by 3:
By dividing both sides by 3, which undoes multiplication, we simplified the equation to \(k = -1\). Understanding inverse operations allows us to manipulate equations systematically.
Variable Term
The "variable term" refers to the part of an equation that includes the variable we are trying to solve for.
It usually appears in the form of a variable (like \(k\), \(x\), or \(y\)) multiplied by a number or between operators.
In our given problem, the variable term is \(3k\). This is because \(k\) is the unknown we want to isolate and solve for.

  • The number 3 in \(3k\) is a coefficient, showing how many times the variable is counted.
  • Our aim is to get \(k\) by itself to determine its value accurately.
Identifying and isolating the variable term is a crucial initial step in any equation-solving process, as it aligns our efforts specifically towards finding the unknown value.

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

The Harvest Nut Company sells made-to-order trail mixes. Sam鈥檚 favorite mix contains peanuts, raisins, and carob-coated pretzels. Peanuts sell for \(\$ 3.20\) per pound, raisins are \(\$ 2.40\) per pound, and the carob-coated sretzels are \(\$ 4.00\) per pound. Sam bought a 5 -pound mixture for \(\$ 16.80\) that contained twice as many pounds of carob-coated pretzels as raisins. How many pounds of peanuts, raisins, and carob-coated pretzels did Sam buy?

Write a system of equations that does not have a unique solution.

Use Cramer鈥檚 Rule to solve each system of equations. \(4 x+2 y-3 z=-32\) \(-x-3 y+z=54\) \(2 y+8 z=78\)

Flight instruction costs \(\$ 105\) per hour, and the simulator costs \(\$ 45\) per hour. Hai-Ling spent 4 more hours in airplane training than in the simulator. If Hai-Ling spent \(\$ 370\) , how much time did he spend training in an airplane and in a simulator?

OPEN ENDED Create a square matrix that does not have an inverse. Explain how you know it has no inverse.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

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.