Problem 8 - Algebra 2

Question

Solve each equation. Check your solution. $$ 3 x=42 $$

Step-by-Step Solution

Answer

The solution is \(x = 14\).

Step 1: Isolate the Variable

To solve the equation \(3x = 42\), you need to find the value of \(x\). Start by isolating \(x\) on one side of the equation. To do this, divide both sides of the equation by 3, the coefficient of \(x\). This gives you: \[x = \frac{42}{3}\]

Step 2: Simplify the Division

Now, perform the division to simplify the equation. Divide 42 by 3 to find the value of \(x\): \[x = 14\]

Step 3: Check the Solution

To verify the solution, substitute \(x = 14\) back into the original equation and ensure the equality holds true. Replace \(x\) in the equation \(3x = 42\) with the solution: \[3(14) = 42\] Compute the product on the left: \[42 = 42\] Since both sides of the equation are equal, the solution \(x = 14\) is correct.

Key Concepts

Isolating the Variable

When solving linear equations, the first step is to isolate the variable you are solving for. A variable is a symbol, often a letter like \(x\), that represents an unknown number we want to determine. In the equation \(3x = 42\), the variable \(x\) is multiplied by 3. To isolate \(x\), perform the inverse operation to undo this multiplication.

This means dividing both sides of the equation by 3. Why divide? Because division is the opposite of multiplication, and using it helps "undo" the multiplication. After dividing the entire equation by 3, the equation becomes \(x = \frac{42}{3}\). Now, \(x\) is alone on one side of the equation. Once the variable is isolated, you can move on to find the actual number it equals.

Equation Simplification

Equation simplification is a crucial part of solving linear equations, helping us to express the equation in its simplest form. In the equation \(x = \frac{42}{3}\), simplification involves performing the division operation. Simplifying \(\frac{42}{3}\) results in \(x = 14\).

Breaking down complex expressions into simpler ones makes it easier to comprehend the solution. Additionally, simplification acts as a checkpoint, showing that every operation performed on the equation is correct. In this context, simplifying helps to clearly identify the value of \(x\).

Simplifying helps us not only reach the correct answer more efficiently but also helps verify each calculation step along the way.

Checking Solutions

Once we have a proposed solution, we need to confirm that it actually satisfies the original equation. This is a straightforward but essential step known as "checking solutions." In our example, after simplifying, we found that \(x = 14\).

To check, substitute \(x = 14\) back into the original equation, \(3x = 42\). When you replace \(x\) with 14, the equation becomes \(3(14) = 42\). Calculating the left side results in\(42\), which matches the right side of the equation, confirming that the solution is correct.

Checking solutions ensures the accuracy of our work. It also provides confidence in our solution method and verifies the correctness of each algebraic manipulation performed earlier.