Problem 20 - Prealgebra

Question

Solve each equation. Check your solution and graph it on a number line. $$19=g-5$$

Step-by-Step Solution

Answer

The solution is \(g = 24\).

Step 1: Isolate the variable

To solve the equation \(19 = g - 5\), we begin by isolating the variable \(g\). This is done by adding 5 to both sides of the equation to balance it. \[19 + 5 = g - 5 + 5\] This simplifies to \[24 = g\].

Step 2: Check the solution

To verify our solution, we substitute \(g = 24\) back into the original equation:\[19 = 24 - 5\]Since this simplifies to \[19 = 19\], our solution is correct.

Step 3: Graph the solution on a number line

We want to plot our solution \(g = 24\) on a number line. Draw a horizontal line and mark a point labelled 24 on it. Since \(g = 24\), place a dot at 24 to represent the value of \(g\).

Key Concepts

Number Line Graph

When solving equations, representing the solution visually can be very helpful. One way to do this is by using a number line graph. A number line is simply a horizontal line marked typically at equal intervals, with numbers written at these marks. This allows you to clearly see where your solution sits in relation to other numbers.

To graph a solution like \(g = 24\) on a number line, you can follow these easy steps:

• Draw a horizontal line and label it with numbers at regular intervals. Make sure these numbers cover the range you'll be plotting.
• Identify the specific number solution, in this case, 24.
• Place a dot or a circle over the point on the number line where 24 is located. This dot represents the solution to the equation.

Number line graphs are especially useful because they provide a visual confirmation of the solution and help in understanding its value in a broader numerical context. This makes equations more tangible and less abstract, especially when working with more complex problems.

Isolation of Variables

In equation solving, one of the first things you'll often want to do is isolate the variable you are solving for. But what does this mean?

Essentially, isolation of variables is the process of manipulating an equation in such a way that you get the variable of interest by itself on one side of the equation. Here are the steps demonstrated using the exercise equation \(19 = g - 5\):

• Start with the original equation: \(19 = g - 5\).
• To get \(g\) by itself, you need to move the 5 that is being subtracted on the right side. This is done by adding 5 to both sides of the equation: \(19 + 5 = g - 5 + 5\).
• When simplifying, we find that \(g = 24\).

This process is crucial because it allows you to focus on the core value you're interested in, helping you solve not just this problem, but providing a foundation for solving any equation effectively. By following these steps, you can handle even more complicated algebraic equations as you progress.

Verification of Solutions

After finding a solution, verifying whether it's correct is an important step. This is called verification of solutions, and ensures that no error has been made in calculations during the solution process.

To verify the solution of an equation, substitute the value you found back into the original equation. If both sides equal, your solution is correct. Let's take a look at how it's done with our exercise:

• We found \(g = 24\). Plug it into the original equation: \(19 = 24 - 5\).
• Calculate the right side: \(24 - 5 = 19\).
• Since both sides of the equation are equal, \(19 = 19\), the solution is verified to be correct.

Verification helps demonstrate that the steps taken were correct and offers confidence in the problem-solving process. This step is especially important in complex scenarios where algebraic manipulation might be tricky, providing a fail-safe against mistakes.