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Chapter 3: Problem 19

Solve each equation. Check your solution and graph it on a number line. $$8=r-5$$

Short Answer

Expert verified
The solution is \(r = 13\). Check confirms it's correct, and on a number line, mark point 13.

Step by step solution

01

Isolate the Variable

To solve the equation, we need to find the value of \(r\) that makes both sides equal. The given equation is \(8 = r - 5\). To isolate \(r\), add 5 to both sides of the equation: \[ 8 + 5 = r - 5 + 5 \] Simplifying both sides gives us: \[ r = 13 \]
02

Check the Solution

With \(r = 13\) as our solution, we substitute back into the original equation to verify:\[ 8 = 13 - 5 \] Calculating the right-hand side gives: \[ 13 - 5 = 8 \]Both sides are equal, confirming that \(r = 13\) is correct.
03

Graph on a Number Line

To represent \(r = 13\) on a number line, draw a horizontal line and place evenly spaced tick marks to indicate integers. Highlight or draw a point above the tick mark labeled '13' to show the solution.

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Key Concepts

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

Isolating the Variable
In the realm of solving linear equations, isolating the variable is a crucial first step. This means rearranging the equation in such a way that the variable you're solving for stands alone on one side. In our case, we started with the equation \(8 = r - 5\). Our goal was to get \(r\) by itself on one side of the equation.

Here's how you do it:
  • Look at the operation around the variable. Here, \( r \) is subtracted by 5.
  • To undo subtraction, add 5 to both sides of the equation. This maintains the equation's balance.
  • Write it out: \(8 + 5 = r - 5 + 5\).
  • Simplify both sides to get \(r = 13\).
You've now isolated the variable, revealing that \(r = 13\). Remember, the goal is to perform inverse operations while maintaining the balance of the equation, just like balancing weights on a scale.
Checking Your Solution
Once you've isolated the variable and found a potential solution, it's important to check your work. This step ensures accuracy and gives confidence that the solution makes sense in the context of the equation.

Here's a straightforward way to check if \(r = 13\) is indeed the solution:
  • Substitute \(r = 13\) back into the original equation: \(8 = 13 - 5\).
  • Perform the subtraction on the right-hand side: \(13 - 5 = 8\).
  • Compare both sides of the equation to confirm that they are equal, \(8 = 8\).
If both sides equal, as they do here, your solution is correct. Checking your solution is key because it helps spot errors and confirms your findings.
Graphing on a Number Line
Visual representation is a powerful tool for understanding and communicating solutions. Graphing the solution on a number line allows students to see where solutions fall in relation to other numbers.

To graph \(r = 13\) on a number line, follow these simple steps:
  • Draw a horizontal line and mark evenly spaced increments.
  • Label the increments with integer values to provide context.
  • Find the position of 13 on your number line.
  • Place a point or circle above the tick mark for 13 to highlight your solution.
This visual representation is especially useful for comparing solutions and understanding the spatial relationships between different numbers. It also reinforces the concept that solutions are real values on the number line.

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