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Chapter 1: Problem 7

Solve the given equations. $$x+5=4$$

Short Answer

Expert verified
The solution to the equation is \( x = -1 \).

Step by step solution

01

Understand the equation

The given equation is \( x + 5 = 4 \). It states that when 5 is added to some number \( x \), the result is 4.
02

Isolate the variable

To find the value of \( x \), we need to get \( x \) by itself on one side of the equation. We can do this by subtracting 5 from both sides of the equation.
03

Perform the subtraction

Subtract 5 from both sides: \( x + 5 - 5 = 4 - 5 \).
04

Simplify both sides

On the left side, \( 5 - 5 \) cancels out, leaving \( x \). On the right side, \( 4 - 5 \) equals \( -1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding and Solving Equations
Solving equations is a fundamental part of basic algebra. At its core, an equation is a statement that asserts the equality of two expressions. When we solve an equation, we search for the value of the variable that makes the equation true.
Consider the equation given: \( x + 5 = 4 \). Our goal is to determine the value of \( x \) that makes the equation hold true.
To solve this equation, we need to manipulate it in such a way that the variable \( x \) is on one side of the equation by itself. This is where the concept of isolating the variable comes into play.
Isolating the Variable
Isolating the variable means rearranging the equation so that the variable, in this case \( x \), stands alone on one side of the equation. This helps us clearly see the solution for the variable.
In the equation \( x + 5 = 4 \), the variable \( x \) is currently combined with the number 5. To isolate \( x \), we must remove the 5 from the left side.
To do so, we can use basic algebraic operations to subtract. This method involves subtracting the same number from both sides of the equation, ensuring the equation remains balanced.
Subtracting from Both Sides of the Equation
Subtracting the same number from both sides of an equation is a common technique used in algebra to simplify equations and isolate variables.
For the equation \( x + 5 = 4 \), we subtract 5 from both sides to maintain equality:
  • Subtract 5 from the left side: \( x + 5 - 5 \)
  • Subtract 5 from the right side: \( 4 - 5 \)
On simplifying, the left side, \( 5 - 5 \), cancels out, leaving only \( x \). Meanwhile, the right side simplifies to \( -1 \).
Thus, we find that \( x = -1 \), completing the process of solving the equation by using subtraction to isolate the variable.

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Most popular questions from this chapter

Find the indicated values. In forestry, a formula used to determine the volume \(V\) of a log is \(V=\frac{1}{2} L(B+b),\) where \(L\) is the length of the log and \(B\) and \(b\) are the areas of the ends. Find \(b\) (in \(\mathrm{ft}^{2}\) ) if \(V=38.6 \mathrm{ft}^{3}, L=16.1 \mathrm{ft}\) and \(B=2.63 \mathrm{ft}^{2} .\) See Fig. 1.17

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