Problem 17 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$k-6=13$$

Step-by-Step Solution

Answer

The solution is \(k = 19\).

Step 1: Isolate the variable

Start by adding 6 to both sides of the equation to isolate the variable \(k\).\[ k - 6 + 6 = 13 + 6 \] This simplifies to:\[ k = 19 \]

Step 2: Check the solution

To ensure the solution is correct, substitute \(k = 19\) back into the original equation. \[ 19 - 6 = 13 \] Since both sides of the equation are equal, \(k = 19\) is the correct solution.

Step 3: Graph on a number line

To graph \(k = 19\) on a number line, draw a number line and place a point at 19. You can create a simple line with numbers and mark the 19th position clearly, indicating that is where the solution lies. Ensure that values like 18 and 20 are visible on either side of 19 for reference.

Key Concepts

Isolate the Variable

To solve an equation, one of the first and most crucial steps is to isolate the variable. This means you want to get the variable all by itself on one side of the equation, usually the left side. When you isolate the variable, you're essentially unraveling the equation to find out what that variable equals.
Here's how you do it:

• Identify the operations being performed on the variable, such as addition, subtraction, multiplication, or division.
• Use inverse operations to cancel those out. In the case of the equation given, \(k - 6 = 13\), subtraction was used, so you'll apply addition.
• Apply these operations equally to both sides of the equation. This keeps the equation balanced.

For example, adding 6 to both sides of \(k - 6 = 13\) simplifies it to \(k = 19\). Now, the variable \(k\) is isolated, revealing its value.

Checking Solutions

Once you've solved an equation by finding the variable's value, it's essential to check your work. This helps you verify that your solution is correct and no mistakes were made during the equation-solving process.
Here's how to check the solution:

• Take the value you found for the variable, in this case, \(k = 19\).
• Substitute this value back into the original equation.
• Perform the operations to see if both sides of the equation remain equal.

For the given problem:- Substitute \(19\) for \(k\) in \(k - 6 = 13\).- Calculate \(19 - 6\) and verify that it indeed equals 13.
If both sides match, your solution is correct! This step is crucial and provides confidence that the variable's value truly satisfies the equation.

Graphing on a Number Line

Graphing the solution on a number line offers a visual representation that can make understanding the solution more intuitive. Even if drawing isn't necessary for all math problems, it can be beneficial.
Here's how to graph a solution:

• Draw a horizontal line and mark it with evenly spaced numbers.
• Identify where your solution falls on this line. In the exercise, this would be \(k = 19\).
• Place a clear point or dot on the number line at 19 to indicate the position of your solution.

For additional clarity:- Make sure neighboring numbers, like 18 and 20, are also labeled. This provides context and ensures viewers though know where 19 is.
Graphing helps solidify your understanding of the solution and offers a practical way to visualize mathematical concepts.