Problem 18 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$r-5=10$$

Step-by-Step Solution

Answer

The solution to the equation is \( r = 15 \).

Step 1: Identify the Equation

The given equation to be solved is \( r - 5 = 10 \). Our goal is to solve for \( r \).

Step 2: Isolate the Variable

To isolate \( r \), add 5 to both sides of the equation. This will cancel out the \(-5\) on the left side. Thus, we have: \[ r - 5 + 5 = 10 + 5 \] which simplifies to \[ r = 15 \].

Step 3: Check the Solution

Substitute \( r = 15 \) back into the original equation to ensure it satisfies the equation. Replace \( r \) in \( r - 5 = 10 \) with 15 to get: \[ 15 - 5 = 10 \]. This simplifies to \( 10 = 10 \), which is accurate, confirming our solution is correct.

Step 4: Graph on a Number Line

Draw a horizontal line, and mark a point labeled 15 on it. This point represents the solution to the equation \( r = 15 \). Since it's not an inequality, there is no need for shading or additional marks.

Key Concepts

Solving Equations

When tackling equations, the goal is to find the value of the variable that makes the equation true. In the exercise given, the equation is \( r - 5 = 10 \). Our mission is to solve for \( r \).

• Isolating the variable means you want \( r \) by itself on one side of the equation.
• To achieve this, perform the opposite operation of what’s currently being applied to \( r \), which in this case, is subtraction.
• Add 5 to both sides of the equation: \( r - 5 + 5 = 10 + 5 \).
• This simplifies to \( r = 15 \).

By following these steps, you effectively isolate the variable, revealing that \( r = 15 \). This straightforward process is crucial in mastering prealgebra equations.

Checking Solutions

To ensure your solution is accurate, check by substituting the found value back into the original equation.

• After solving \( r = 15 \), substitute 15 back into the original equation: \( r - 5 = 10 \).
• Replace \( r \) with 15: \( 15 - 5 = 10 \).
• Simplify: \( 10 = 10 \).

This confirms that both sides of the equation are equal, validating that your solution \( r = 15 \) is correct.
Checking solutions not only builds confidence in your answer but reinforces the understanding of how algebraic operations maintain equality.

Graphing on a Number Line

Graphing the solution on a number line gives a visual representation of the answer.

• Draw a horizontal line which will serve as your number line.
• Identify a point corresponding to the solution, in this case, 15.
• Mark the point clearly on the number line.

For the equation \( r = 15 \), simply marking the point suffices since it isn't an inequality.
Graphing is a tool to visually confirm the solution and enhances understanding of the solution's value, ensuring a comprehensive grasp on the concept of solving equations.