Problem 57 - Prealgebra

Question

Solve each equation. Check your solution. $$5=10 a$$

Step-by-Step Solution

Answer

The solution is \(a = \frac{1}{2}\).

Step 1: Isolate the Variable

The given equation is \(5 = 10a\). To solve for \(a\), we need to isolate \(a\) on one side of the equation. Since \(10a\) means 10 times \(a\), we can isolate \(a\) by dividing both sides of the equation by 10. This gives us: \[ a = \frac{5}{10} \]

Step 2: Simplify the Fraction

Now simplify the fraction \(\frac{5}{10}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 5. So, we get: \[ a = \frac{5 \div 5}{10 \div 5} = \frac{1}{2} \]

Step 3: Verify the Solution

To check if \(a = \frac{1}{2}\) is correct, substitute \(a\) in the original equation: \(5 = 10a\). Substitute: \[ 5 = 10 \times \frac{1}{2} \] Calculate the right side: \[ 10 \times \frac{1}{2} = 5 \] Since both sides are equal, the solution is verified.

Key Concepts

Isolating Variables

The process of isolating variables is essential when solving equations, as it allows you to focus on finding the value of the unknown variable. For the equation given in the original exercise, you start with

• \( 5 = 10a \)

Our goal is to get 'a' by itself on one side of the equation. This means eliminating everything else that is multiplied by, divided by, or added to 'a'. Since we have multiplication involved—specifically, '10 times a'—we perform the opposite operation to eliminate it.
To isolate 'a', divide both sides of the equation by 10, the coefficient of 'a'. By doing this, you keep the equation balanced. When you divide,

• \( 10a \) by 10, you get 'a'.
• Similarly, \( 5 \) divided by 10 gives you \( \frac{5}{10} \).

This results in:

• \( a = \frac{5}{10} \).

Now, 'a' is isolated, and you can see its relationship to the numbers on the other side. Isolating the variable is a key step before any further simplification or solution validation.

Checking Solutions

After isolating the variable and solving the equation, it is crucial to verify that the solution is correct. This involves plugging your found value back into the original equation.
This ensures that our solution checks out in the context of the problem. From the solution provided, we had reached:

• \( a = \frac{1}{2} \)

To verify, substitute \( \frac{1}{2} \) back into the original equation:

• \( 5 = 10a \)

Replace 'a' with \( \frac{1}{2} \):

• \( 5 = 10 \times \frac{1}{2} \)

Perform the multiplication on the right side:

• \( 10 \times \frac{1}{2} = 5 \)

You see that the left side (5) equals the right side (5). This confirms the solution's validity. Checking your solutions is a good habit to ensure your work is accurate and to gain confidence in your mathematical abilities.

Simplifying Fractions

Simplifying fractions reduces them to their simplest form, making them easier to understand and work with. This process involves finding the greatest common divisor (GCD) of the numerator and the denominator. In our exercise, after isolating the variable, we were left with the fraction

• \( \frac{5}{10} \).

The greatest common divisor of 5 and 10 is 5. Simplify the fraction by dividing both the numerator and the denominator by their GCD:

• Divide the numerator (5) by 5 to get 1.
• Divide the denominator (10) by 5 to get 2.

This results in the simplified fraction:

• \( \frac{1}{2} \).

Simplifying fractions is an integral step in solving equations neatly and is a valuable skill for making further calculations more manageable. Understanding how to simplify can help you verify your solutions quickly and enhance clarity in your work.