Problem 13 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 7 x=42 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 6 \).

Step 1: Identify the Equation

The given equation is \( 7x = 42 \). This is a simple linear equation in one variable.

Step 2: Isolate the Variable

To solve for \( x \), we need to isolate it. We can do this by dividing both sides of the equation by 7. This gives us \( x = \frac{42}{7} \).

Step 3: Perform the Division

Divide 42 by 7, which results in \( x = 6 \).

Step 4: Check the Result

Substitute \( x = 6 \) back into the original equation to verify the solution: \( 7(6) = 42 \). This simplifies to \( 42 = 42 \), confirming that our solution is correct.

Key Concepts

Solving Linear Equations

Linear equations are among the most fundamental concepts in algebra. They involve equations where the highest power of a variable is one. To solve linear equations, you have to find the value of the variable that makes the equation true. These equations are simple yet essential building blocks for more complicated mathematics.

In the exercise we are looking at, we have the equation \(7x = 42\). Here, "7x" means 7 times some number \(x\) and it equals 42. Our goal is to find the value of \(x\) that makes this equation true. This is done by converting the equation into an easy-to-understand form, with the variable isolated on one side. By doing so, solving the equation becomes straightforward and logical.

Variable Isolation

Variable isolation is a key step in solving any linear equation. It means getting the variable on its own on one side of the equation without any additional numbers or terms attached to it.

For \(7x = 42\), isolating \(x\) involves performing operations that simplify the equation to give us \(x = ...\). Here’s how it works:

• We start by removing the multiplication involving \(x\). Since \(7x\) means 7 times \(x\), we do the opposite of multiplication, which is division.
• By dividing both sides of the equation by 7, we effectively cancel out the 7 on the left side.

This step gives us \(x = \frac{42}{7}\), which simplifies to \(x = 6\). Now, the variable \(x\) is isolated and ready for checking.

Equation Checking

After solving the equation, it's important to check your solution. Checking ensures that the answer is correct and confirms that no mistakes were made during the calculations.For our equation, checking simply involves substituting the value we found for \(x\) back into the original equation:

• We found \(x = 6\), so substitute \(6\) back into \(7x = 42\) to see if the equation holds true.
  This becomes \(7(6)\), which simplifies to \(42\).
• If the left side equals the right side (\(42 = 42\)), your solution is correct.

Equation checking confirms that our computations were accurate, and the value of \(x\) indeed satisfies the original equation.