Problem 15 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 10 x=120 $$

Step-by-Step Solution

Answer

The solution for the equation is \( x = 12 \).

Step 1: Understand the Equation

The equation given is a linear equation: \( 10x = 120 \). Our goal is to find the value of \( x \) that satisfies this equation.

Step 2: Isolate the Variable

To solve for \( x \), we need to isolate it on one side of the equation. Currently, \( x \) is multiplied by 10. To isolate \( x \), we need to perform the opposite operation, which is division. Therefore, divide both sides of the equation by 10.

Step 3: Perform the Division

Divide both sides of the equation by 10 to isolate \( x \).\[ \frac{10x}{10} = \frac{120}{10} \]This simplification gives us:\[ x = 12 \]

Step 4: Check the Solution

Substitute \( x = 12 \) back into the original equation to verify the solution. Original equation: \( 10x = 120 \)Substitute \( x \) with 12:\[ 10 \times 12 = 120 \]Since both sides of the equation are equal, our solution is verified as correct.

Key Concepts

Isolation of Variables

In linear algebra, solving equations usually involves simplifying the equation to find the value of a variable. One key technique employed is isolation of the variable. The idea here is to get the variable you are solving for by itself on one side of the equation. This method is crucial in finding what value the variable holds.

Let's look at the equation from the exercise: \(10x = 120\). The goal is to have \(x\) by itself on one side. Here, \(x\) is currently multiplied by 10. To isolate \(x\), you need to do the opposite operation of multiplication, which is division.

You divide both sides of the equation by 10. Here's the process described in simple steps:

• Perform the opposite mathematical operation: Since \(x\) is multiplied, divide both sides by 10.
• Maintain equality: Always remember to do the same operation on both sides to keep the equation balanced.

After performing this step, you get \(x = 12\), showing that the variable is isolated successfully.

Substitution Method

Once you've solved an equation through isolation of variables and found a proposed solution, it often helps to use the substitution method to verify that this proposed solution is correct. This involves taking the solution you have found and substituting it back into the original equation to see if it holds true.

In our example, we found \(x = 12\). Using the substitution method involves replacing \(x\) in the original equation \(10x = 120\) with 12. The steps are straightforward:

• Substitute the variable: Replace \(x\) with 12 in the original equation.
• Solve the equation: Perform the multiplication \(10 \times 12\).
• Check that the equation balances: Both sides should be equal for the solution to be correct.

When you execute this substitution, \(10 \times 12 = 120\), confirming that the solution \(x = 12\) is indeed correct. This method is a reliable way to ensure your solution is accurate.

Checking Solutions

Checking your solutions is the final and a very important step in solving equations. It ensures that the value you obtained for the variable actually satisfies the equation and isn't a result of simple calculation errors.

Here's how to efficiently check solutions:

• Re-substitute into the original equation: Just like in the substitution method, use the value of the variable you found.
• Calculate both sides: Perform the necessary operations to confirm both sides are equal.
• Verify: Make sure that both sides of the equation match exactly. This verifies your solution.

In our linear equation example, substituting \(x = 12\) gives \(10 \times 12 = 120\). Both sides equal 120, confirming that the equation is true and the solution is correct. Always remember to check solutions to avoid mistakes and clarify doubts about your answers.