Problem 6 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$r-4=-18$$

Step-by-Step Solution

Answer

The solution to the equation is \( r = -14 \).

Step 1: Isolate the Variable

We want to solve the equation \( r - 4 = -18 \) by isolating \( r \). To do this, add 4 to both sides of the equation:\[r - 4 + 4 = -18 + 4\]This simplifies to:\[r = -14\]

Step 2: Verify the Solution

To check if \( r = -14 \) is correct, substitute \( -14 \) back into the original equation and see if both sides are equal:\[(-14) - 4 = -18\]Calculate the left side: \[-18 = -18\]Both sides are equal, verifying that \( r = -14 \) is indeed a solution.

Step 3: Graph on a Number Line

To represent \( r = -14 \) on a number line:1. Draw a horizontal line.2. Mark a point at \(-14\) on this line.3. Place a filled circle on the number \(-14\) to indicate that this is the solution of the equation. Your number line should show this point clearly.

Key Concepts

Check Solutions

When solving a linear equation, it's crucial to check if your answer is correct. This process ensures that an error wasn't made during calculation.
To check a solution, substitute the value you found for the variable back into the original equation.
- If both sides of the equation are equal after substitution, the solution is accurate. - If they're not equal, it means there's a mistake in your calculations.For example, in the equation \( r - 4 = -18 \), we found that \( r = -14 \). Substituting \( -14 \) back into the equation provides \((-14) - 4 = -18\), verifying our solution is correct. Therefore, checking solutions is a simple but vital step in solving equations to ensure accuracy and understanding.

Graphing on a Number Line

Graphing solutions on a number line is a visual representation that helps in understanding where the solution lies in relation to other numbers.
Here's how you can graph a solution on a number line:- Draw a horizontal line that serves as your number line.- Identify the solution and locate it appropriately on the line. - For our equation, we identified the solution as \( r = -14 \). Find \(-14\) on the number line.- Place a filled circle on \(-14\) to indicate that the solution is included and exact.The circle is filled because \( r = -14 \) is exactly the solution we're graphing. This representation confirms the problem is solved correctly and aids in visual comprehension.

Isolation of Variables

Isolation of variables is the technique used to solve an equation by getting the variable alone on one side of the equation. This process uncovers the value of the variable that makes the equation true.
To isolate a variable, perform operations that cancel out other numbers either by addition, subtraction, multiplication, or division.For instance, in the equation \( r - 4 = -18 \), the goal is to make \( r \) (the unknown) stand alone. - Since 4 is subtracted from \( r \), add 4 to both sides to cancel the \(-4\).This results in:\[ r - 4 + 4 = -18 + 4 \]Simplifying the equation gives:\[ r = -14 \]Now, \( r \) is isolated. The variable is on one side by itself, and its value is known, which completes solving the equation. Understanding isolation of variables equips students with the skills to tackle a variety of algebraic problems effectively.