/*! 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 69 The sum of \(4 x\) and 3 is equa... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 69

The sum of \(4 x\) and 3 is equal to the difference of \(7 x\) and 8 .

Short Answer

Expert verified
The value of \(x\) is \(\frac{11}{3}\).

Step by step solution

01

Setting up the Equation

Begin by writing the equation according to the problem statement. The sum of \(4x\) and 3 is the same as the difference between \(7x\) and 8. Thus, the equation is: \(4x + 3 = 7x - 8\).
02

Isolating Variable Terms

To solve for \(x\), you first want to get all \(x\) terms on one side. Subtract \(4x\) from both sides: \(3 = 7x - 4x - 8\). This simplifies to \(3 = 3x - 8\).
03

Isolating the Constant Term

Next, isolate the \(3x\) term by adding 8 to both sides of the equation: \(3 + 8 = 3x\). This simplifies to \(11 = 3x\).
04

Solving for x

To find the value of \(x\), divide both sides of the equation by 3: \(x = \frac{11}{3}\).

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.

Linear Equations
Linear equations are a fundamental concept in algebra. They involve any equation that can be written in the form \(Ax + B = C\), where \(A\), \(B\), and \(C\) are constants. In this exercise, the equation is set as \(4x + 3 = 7x - 8\).
Linear equations represent a straight line when graphed on a Cartesian plane. Solving a linear equation typically involves finding the value of the variable that makes the equation true. Such problems are among the building blocks of algebra, helping us understand more complex math concepts later on.
Variable Isolation
Variable isolation is about getting the variable, in this case \(x\), alone on one side of the equation. This allows us to determine its value. Start by moving all terms containing \(x\) to one side and constants to the other.
In our problem, the equation \(4x + 3 = 7x - 8\) was reorganized by subtracting \(4x\) from both sides: \(3 = 7x - 4x - 8\). This leaves \(3x\) on the right side, isolating the variable term. Doing this simplifies the process and brings the problem to a point where you can solve for \(x\) easily.
Algebraic Manipulation
Algebraic manipulation involves rearranging the equation using various algebraic operations like addition, subtraction, multiplication, and division to make it easier to solve.
After isolating \(x\) terms, we need to handle constants next by adding or subtracting them as necessary. Here, we added 8 to both sides of \(3 = 3x - 8\) which resulted in \(11 = 3x\). This form of manipulation makes the equation clearer and paves the way to solve for the unknown variable.
Problem Solving Steps
Problem solving in math is a sequence of logical steps. Here鈥檚 how it unfolds for this exercise:
  • Step 1: Write the equation according to the problem statement.
  • Step 2: Simplify it by isolating the variable terms.
  • Step 3: Further simplify by isolating the constant so it's ready to solve.
  • Step 4: Solve for \(x\) using division.

Each step builds upon the last, making the solution process clear. Following a structured approach helps prevent errors and ensures you solve linear equations effectively.

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

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ x+7 \leq 5 \text { and } x-3 \geq-10 $$

A recipe calls for \(11 / 2\) teaspoons of vanilla extract for every 3 cups of batter. How many teaspoons of vanilla extract should be used with 7 cups of batter?

Solve and graph. In addition, present the solution set in interval notation. $$ x+2>-1 $$

Solve and graph. In addition, present the solution set in interval notation. $$ 9 x+4 \leq-5 $$

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -2 \leq 12-2(x-3) \leq 20 $$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Mechanics 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.