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Chapter 4: Problem 18

Solve each equation. $$x-8=-3$$

Short Answer

Expert verified
The solution is \(x = 5\).

Step by step solution

01

Identify the Objective

We need to find the value of \(x\) that makes the equation \(x - 8 = -3\) true.
02

Add 8 to Both Sides

To isolate \(x\), we should eliminate the \(-8\) on the left side by adding 8 to both sides of the equation. This yields: \(x - 8 + 8 = -3 + 8\).
03

Simplify both Sides

Simplify both sides of the equation. The left side simplifies to \(x\), and the right side simplifies to \(5\), so the equation becomes \(x = 5\).

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Key Concepts

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

Steps for Solving Equations
When it comes to solving linear equations, following a systematic approach can simplify the task. Clear steps for solving equations not only build a stronger foundation in mathematics but also make the process straightforward and organized. Here is how you should approach it:
  • Understand the Problem: Begin by examining what the equation is asking you to do. Identify the variable you need to solve for and the operations involved.
  • Isolate the Variable: The primary goal is to have the variable on one side of the equation. This often entails performing operations such as addition, subtraction, multiplication, or division.
  • Check Your Solution: After finding the value of the variable, always substitute it back into the original equation to ensure that it satisfies the equation.
By following these steps, you can effectively tackle any basic linear equation.
Isolating the Variable
In any linear equation, isolating the variable is a key step. It means getting the variable by itself on one side of the equation. This process involves reversing the operations around the variable. Let's use the equation given: \(x - 8 = -3\).To isolate \(x\), we need to get rid of the \(-8\). Since \(-8\) is subtracted from \(x\), we do the opposite operation鈥攁ddition. Therefore, we add 8 to both sides:
  • Initial equation: \(x - 8 = -3\)
  • Add 8 to both sides: \(x - 8 + 8 = -3 + 8\)
This process brings us a step closer to pinpointing the value of the variable. Remember, whatever you do to one side of the equation, you must do to the other side to maintain balance.
Simplifying Equations
Once you've isolated the variable, the next logical step is simplifying the equation. This is where you tidy up and make sure everything is clear and concise. For the equation we are working with, simplifying means performing the arithmetic operations to eliminate any redundant terms:
  • After adding 8 to both sides, simplify: \(x - 8 + 8\) simplifies to \(x\), and \(-3 + 8\) simplifies to \(5\).
  • The equation thus becomes: \(x = 5\).
Simplification ensures that the equation is easy to interpret and that your solution is clear. It's important to double-check that every operation has been correctly carried out and that any unnecessary terms are removed. This attention to detail guarantees a clean and correct result.

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Most popular questions from this chapter

Using the addition property of equality first, solve each of the following equations. $$2 x+4=6$$

One angle in a triangle is half the largest angle, but three times the smallest. Find all three angles.

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$7 x=-35$$

Find the complement and supplement of each angle. [Example \(6]\) $$59^{\circ}$$

Solve each equation using the methods shown in this section. $$2(3 x-6)+1=7$$
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