Problem 49 - Algebra A Combined Function

Question

\(42=7 x\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Equation

The given equation is \(42 = 7x\). This is a simple linear equation where you need to solve for \(x\). It means that \(7\) times some number \(x\) equals \(42\).

Step 2: Isolate the Variable

To find the value of \(x\), you need to isolate it on one side of the equation. To do this, divide both sides of the equation by \(7\). This gives you: \[\frac{42}{7} = \frac{7x}{7}\]

Step 3: Solve for \(x\)

Now, simplify both sides. \(\frac{42}{7} = 6\), and \(\frac{7x}{7} = x\). Thus, \(x = 6\).

Step 4: Verify the Solution

To ensure our solution is correct, substitute \(x = 6\) back into the original equation: \(42 = 7 \times 6\). Calculating the right side gives \(42\), which matches the left side, confirming that our solution is correct.

Key Concepts

Solving for a Variable

Solving for a variable in a linear equation means finding the value of the variable that makes the equation true. For example, in the equation \(42 = 7x\), our goal is to find what number \(x\) must be so that when multiplied by 7, the result equals 42. This process is crucial because it helps us understand the relationship between numbers and allows us to deduce unknown quantities from known information. Linear equations typically have one solution, and solving easily provides us with this value.
To solve an equation, start by determining what operation needs to be reversed to isolate the variable. This might involve addition, subtraction, multiplication, or division, depending on how the equation is structured. Always perform the same operation on both sides of the equation to maintain equality. This is the fundamental principle of solving equations—what you do to one side, you must do to the other.

Isolating the Variable

Isolating the variable involves arranging the equation so that the variable stands alone on one side, usually the left, of the equation. It's like peeling away layers that obscure the true value of the variable. Let's take the equation \(42 = 7x\). To isolate \(x\), you need to remove the 7 that is multiplying it.
Since multiplication is the operation joining 7 and \(x\), do the opposite: divide. By dividing both sides by 7, you isolate \(x\):

• \(\frac{42}{7} = \frac{7x}{7}\)
• Simplifying this gives \(6 = x\)

The process of isolating a variable follows the basic property that you can do anything to an equation as long as you do the same thing to both sides. It's about balancing your actions to reveal the value of the variable.

Verifying Solutions

Verifying your solution ensures that the value you found for the variable truly satisfies the equation. It's like a final check to confirm your answer is correct. For the solved equation \(42 = 7x\), after isolating \(x\) and solving, we found \(x = 6\). To verify:1. Substitute \(x = 6\) back into the original equation.

• Replace \(x\) with 6: \(42 = 7 \times 6\)
• Simplify the right side: \(42 = 42\)

2. Since both sides match, our solution is verified. This step boosts your confidence because it confirms that all operations were performed accurately.Verification is essential because it exposes any mistakes made during the earlier steps, allowing correction and understanding before moving forward. It's always good practice to verify solutions in mathematical problems.