/*! 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 77 For exercises 65-86, (a) solve... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 77

For exercises 65-86, (a) solve. (b) check. $$ 15 p-(8 p+2)=3(p+2)+4(p-1) $$

Short Answer

Expert verified
No solution.

Step by step solution

01

Simplify Both Sides

Begin by simplifying both sides of the equation. Expand the expressions: Left Side: 15p - (8p + 2) = 15p - 8p - 2 = 7p - 2 Right Side: 3(p + 2) + 4(p - 1) = 3p + 6 + 4p - 4 = 7p + 2
02

Set Simplified Expressions Equal

Now that the expressions are simplified, set them equal to each other: 7p - 2 = 7p + 2
03

Solve for p

To solve for p, subtract 7p from both sides: 7p - 2 - 7p = 7p + 2 - 7p This simplifies to: -2 = 2 Since this statement is false, it means there is no solution for p.
04

Check the Solution

We've already found that there is no solution, so there is no need to check further. The equation is inconsistent, meaning there is no value for p that will satisfy it.

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.

simplifying expressions
To solve linear equations, it's crucial to simplify expressions first. Simplifying involves combining like terms and expanding any parentheses on both sides of the equation.
In our example, we have:
Left Side: 15p - (8p + 2).
We distribute the minus sign: 15p - 8p - 2 = 7p - 2.
On the Right Side: 3(p + 2) + 4(p - 1).
Distribute the terms inside the parentheses: 3p + 6 + 4p - 4 = 7p + 2.
Simplifying these expressions will make solving the equation easier.
Always remember these steps:
  • Expand parentheses
  • Combine like terms
checking solutions
After simplifying and solving an equation, it's important to check your solution. This ensures your solution is correct.
In our problem, we simplify to find no solution. However, for typical cases, you substitute the solution back into the original equation to verify it holds true. Follow these steps:
  • Insert the found value back into the original equation
  • Simplify both sides of the equation
  • Ensure both sides are equal
If they are equal, your solution is correct. If not, you should recheck your steps.
inconsistent equations
Inconsistent equations are equations that have no solution. This means no value can satisfy the equation.
In our example, after simplifying, we found:
7p - 2 = 7p + 2
After canceling out terms, we get:
-2 = 2
This is a false statement, indicating our equation is inconsistent.
Recognizing inconsistent equations is key in algebra. They typically result from contradictions in the equation after simplification.
no solution
When solving equations, you might sometimes find there is no solution. This means the equation cannot be true no matter what value is substituted in.
For our problem, the simplified forms led to a false statement, showing -2 equals 2. Since this is impossible, the conclusion is there is no solution.
In general, when you simplify both sides of an equation and get a contradiction, there is no solution.
Remember:
  • If simplifying leads to a false statement (like -2 = 2), the equation has no solution
  • This means the initial equation is inconsistent

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

For exercises 77-78, \(V=\frac{L W T}{144 \text { in. }^{3}}\) where \(V\) is the volume of lumber in board feet, \(L\) is the length, \(W\) is the width, and \(T\) is the thickness. The units of length, width, and thickness are inches. a. Solve \(V=\frac{L W T}{144 \text { in. }^{3}}\) for \(L\). Include the units of measurement. b. The thickness of each board in a pile is \(1.5\) in., and the width of each board is \(3.5\) in. The volume of the boards is 182 board feet. Find the total length of the boards. c. Change this length into feet.

For exercises \(65-72\), assign a variable, and write an inequality that represents the constraint. An employee is working a temporary job for \(\$ 9\) per hour. Her employer withholds \(7.65 \%\) of her wages to pay for Social Security and Medicare. Her rent payment is \(\$ 420\) per month. Find the number of hours the employee must work to earn enough to at least pay her next two rent payments.

For exercises 37-52, (a) solve. (b) use a number line graph to represent the solution. (c) check the direction of the inequality sign. $$ -x+7 \leq-15 $$

For exercises 93-96, the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Solve: \(-2 x+12>36\) Incorrect Answer: \(-2 x+12>36\) $$ \begin{aligned} -12 &-12 \\ \hline-2 x+0 &>24 \\ \frac{-2 x}{-2} &>\frac{24}{-2} \\ x &>-12 \end{aligned} $$

For problems \(89-92\), do the arithmetic with a calculator. The area \(A\) of a rectangular playground is \(28,800 \mathrm{ft}^{2}\). The length \(L\) is \(180 \mathrm{ft}\). Solve the formula \(A=L W\) for \(L\), and use it to find the width \(W\) of the playground.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Calculus

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.