/*! 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 6 Solve and check each linear equa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 6

Solve and check each linear equation. $$3 x+5=2 x+13$$

Short Answer

Expert verified
The solution for the equation is \(x = 8\)

Step by step solution

01

Isolate Variable Terms

Firstly, remove the variable term from one side to have the variable on one side of the equation. This is done by subtracting \(2x\) from both sides: \(3x - 2x + 5 = 2x - 2x + 13\). Simplify to get \(x + 5 = 13\)
02

Solve for the Variable

Subtract 5 from both sides of the equation to isolate the variable: \(x + 5 - 5 = 13 - 5\). After simplifying, this results in \(x = 8\)
03

Check the Solution

Finally, substitute \(x = 8\) back into the original equation to double check if it is a correct solution: \(3*8 + 5 = 2*8 + 13\). The left side equals 29 and the right side equals 29 also, confirming that \(x = 8\) is indeed the correct solution

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 Variable
When working with linear equations, one of the first steps is to isolate the variable. This means rearranging the equation so that the variable you're solving for is on one side by itself. In the given equation \(3x + 5 = 2x + 13\), our goal is to get all terms involving \(x\) on one side and constants on the other.
To do this, you can use basic arithmetic operations like addition or subtraction on both sides of the equation. For instance, we subtract \(2x\) from both sides, leading to \(x + 5 = 13\). This step simplifies the equation, making it easier to see what \(x\) equals.
  • Tip: Always perform the same operation on both sides to maintain the balance of the equation.
  • This step sets the stage for solving the equation by clearly showing what operations need to be further performed on the isolated term.
Substitution
Substitution is the step where you verify your solution by plugging it back into the original equation. After determining that \(x = 8\) from the equation \(x + 5 = 13\), you substitute \(8\) for \(x\) in the original equation \(3x + 5 = 2x + 13\).
This means calculating \(3 \times 8 + 5\) and \(2 \times 8 + 13\). Both sides should be equal if \(x = 8\) is indeed the correct solution. Thus, you find both the left and right side equals \(29\), confirming our solution鈥檚 validity.
  • This step is crucial because it helps to catch any potential mistakes made in the earlier steps by providing a clear confirmation.
  • Always consider substitution as a final safety check to ensure that your solution is not just plausible, but correct.
Simplify Algebraic Expressions
Simplifying algebraic expressions is an essential part of solving equations. It involves reducing equations to their simplest form by combining like terms and performing arithmetic operations.
In our step-by-step solution, simplifying played a role when we transformed \(3x - 2x + 5\) into \(x + 5\). Here, \(3x - 2x\) simplifies to \(x\), which is more straightforward to work with.
Additionally, to solve \(x + 5 = 13\), we subtract \(5\) from both sides, which is another aspect of simplification. This results in \(x = 8\), presenting the simplest and most direct form of the equation.
  • By simplifying expressions, equations become easier to handle and less prone to errors.
  • Remember, simplification not only helps in finding the solution quicker but also makes checking work like substitution more manageable.

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

In Exercises 122鈥�133, use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. You are choosing between two texting plans. Plan A has a monthly fee of 15 dollar with a charge of 0.08 dollar per text. Plan \(B\) has a monthly fee of 3 dollar with a charge of 0.12 dollar per text. How many text messages in a month make plan A the better deal?

In Exercises 95鈥�102, use interval notation to represent all values of x satisfying the given conditions. \(y=1-(x+3)+2 x\) and \(y\) is at least 4

List all numbers that must be excluded from the domain of each rational expression. $$ \frac{3}{2 x^{2}+4 x-9} $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) $$ \begin{aligned} 2 &>1 \\ 2(y-x) &>1(y-x) \\ 2 y-2 x &>y-x \\ y-2 x &>-x \\ y &>x \end{aligned} $$ The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\)

List all numbers that must be excluded from the domain of each rational expression. $$ \frac{7}{2 x^{2}-8 x+5} $$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.