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Chapter 3: Problem 76

\(x-5=0\)

Short Answer

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

Step by step solution

01

Understand the Equation

We have the equation: \[ x - 5 = 0 \]Our goal is to solve for the value of \(x\) that makes this equation true.
02

Isolate \(x\)

To isolate \(x\), we need to move the constant term (-5) to the other side of the equation. To do this, we add 5 to both sides of the equation.
03

Perform the Addition

Add 5 to both sides of the equation: \[ x - 5 + 5 = 0 + 5 \]This simplifies to: \[ x = 5 \]
04

Verify the Solution

To ensure the solution is correct, substitute \(x = 5\) back into the original equation: \[ 5 - 5 = 0 \]This is a true statement, so \(x = 5\) is the correct solution.

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Key Concepts

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

Isolate the Variable
When solving a linear equation, the main goal is to isolate the unknown variable on one side of the equation. In our example, the equation is \[ x - 5 = 0 \]. To isolate the variable x, follow these steps:
  • Identify the constant term that is being subtracted or added to the variable. In this case, the constant term is -5.
  • Move the constant term to the other side of the equation. You can do this by performing the opposite operation. Here, we add 5 to both sides of the equation to counteract the -5:

    • \[ x - 5 + 5 = 0 + 5 \]
  • When you add or subtract the same number on both sides of an equation, the equality is maintained. This simplifies our equation to:
    • \[ x = 5 \]
Now, the variable x is isolated, and we have found our solution!
Verify the Solution
After isolating the variable and finding a solution, it is crucial to verify that the solution is correct. Verification ensures that you have performed the steps accurately. For our example equation \[ x - 5 = 0 \] we determined that the solution is \[ x = 5 \]. To verify this solution, substitute x = 5 back into the original equation:
  • Replace x in the equation with 5:

    • \[ 5 - 5 = 0 \]
  • Simplify the expression on the left-hand side:

    • \[ 0 = 0 \]
This statement is true, verifying that our solution \[ x = 5 \] is correct. Verification is a valuable habit that helps prevent errors and reinforces your understanding of the solution.
Simplify the Equation
Simplifying the equation is a crucial step in solving linear equations and typically involves performing operations to both sides of the equation to make it more manageable. In our example, the original equation is: \[ x - 5 = 0 \] To simplify:
  • Identify any like terms or constants that can be combined or moved. In this case, we want to move the -5.
  • Perform the necessary operation on both sides of the equation to keep it balanced. Here, we add 5 to both sides:

    • \[ x - 5 + 5 = 0 + 5 \]
  • After performing the operation, simplify the terms to isolate the variable:

    • \[ x = 5 \]
Simplifying helps in reducing complexity and makes it easier to identify the variable's value, effectively solving the equation.

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Most popular questions from this chapter

$$ y=\frac{5}{2} x-4 $$

\(\begin{aligned} 5 x-y &=1 \\ x-5 y &=-10 \end{aligned}\)

Graph each linear equation. $$ y=-6 x $$

Plot each set of points, and draw a line through them. Then give an equation of the line. $$ (-5,5),(0,5) \text { and }(3,5) $$

Find the \(x\) - and \(y\) -intercepts for the graph of each equation. $$ y=-1.5 $$
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