Problem 10 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$x+5=18$$

Step-by-Step Solution

Answer

The solution is \(x = 13\).

Step 1: Isolate the Variable

Start with the equation \(x + 5 = 18\). To isolate \(x\), subtract 5 from both sides of the equation: \(x + 5 - 5 = 18 - 5\). This simplifies to \(x = 13\).

Step 2: Check the Solution

Substitute \(x = 13\) back into the original equation to verify it solves the equation: \(13 + 5 = 18\). Since both sides equal 18, the solution \(x = 13\) is correct.

Step 3: Graph on the Number Line

To graph \(x = 13\) on a number line, draw a number line and place a point at 13. This represents the solution to the equation.

Key Concepts

Isolating the Variable

When solving linear equations like \( x + 5 = 18 \), the first step is to isolate the variable—here, it's \( x \). This means rearranging the equation so that \( x \) is by itself on one side of the equation. By doing this, you make it easier to see what value \( x \) must be to balance out the equation. To achieve this, you will need to perform operations that "undo" whatever is being done to \( x \) in the equation.Let's break it down:

• Start with the original equation: \( x + 5 = 18 \).
• To isolate \( x \), subtract 5 from both sides: \( x + 5 - 5 = 18 - 5 \).
• This simplifies to \( x = 13 \).

This process of isolating the variable helps to identify that \( x \) equals 13. It’s essential to perform the same operation on both sides of the equation to maintain the balance, just like balancing a scale.

Checking the Solution

Once you have isolated the variable and found that \( x = 13 \), it's crucial to ensure that this solution is accurate. This step is known as "checking the solution." By substituting \( x = 13 \) back into the original equation, you can verify that your answer is correct.Here's how you check:

• Substitute \( 13 \) for \( x \) in the original equation: \( 13 + 5 = 18 \).
• Calculate both sides: the left side becomes \( 18 \), which matches the right side of the equation.
• The equality holds true, confirming \( x = 13 \) is indeed the correct solution.

Checking the solution offers confidence in your work and ensures that a small mistake hasn't altered the outcome. It's a safety net that helps verify the accuracy of your calculations.

Graphing on a Number Line

After determining and verifying the solution, the final step is to graph the solution on a number line. This visual representation can help reinforce your understanding of where the solution lies in relation to other numbers.How to graph \( x = 13 \) on a number line:

• Draw a horizontal line and mark evenly spaced numbers along it, which include the number 13.
• Find 13 on the number line and place a point (or a small dot) directly above it to represent the solution \( x = 13 \).

Using a number line as a tool helps in reinforcing the concept of the solution being a single point within the continuum of numbers. It’s a helpful way to visualize solutions, especially when dealing with more complex equations or inequalities.