Problem 36 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$24-5 x=x$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 4\)

Step 1: Simplify the Equation

Combine the terms that contain \(x\) to simplify the equation. This gives us \(24-5x = x\). By adding \(5x\) to both sides, we have \(24 = 6x\).

Step 2: Solve for x

Now that we've isolated 6x on one side, we can solve for \(x\) by dividing both sides of the equation by 6 which gives \(x = 24/6 = 4\).

Step 3: Check the Solution

Substitute \(x = 4\) back into the original equation to check the solution. The left-hand side of the equation is \(24-5*4 = 4\) and the right-hand side is \(x = 4\). Since the left side equals the right side, the solution is correct.

Key Concepts

Equation Simplification

When solving a linear equation, the first goal is to simplify it to make it easier to work with. Consider the equation given:

• \( 24 - 5x = x \)

We need to rearrange the terms containing the variable \(x\) as well as the constants. Start by adding \(5x\) to both sides of the equation. This helps in moving all \(x\) terms to one side, giving us:

• \( 24 = 6x \)

To perform equation simplification effectively, follow these strategies:

• Add or subtract terms to both sides to combine like terms.
• Ensure your variables are on one side and constants on the opposite.
• Continue simplifying until your equation takes its simplest form, with one variable term and one constant term.

This process of simplification lays the groundwork for the next steps in solving linear equations.

Variable Isolation

After simplifying the equation, our next focus turns to isolating the variable \(x\). From the simplified equation:

• \( 24 = 6x \)

Here, \(x\) is multiplied by 6, so we need to isolate \(x\) by performing actions that "undo" this multiplication. Divide every term in the equation by 6:

• \( x = \frac{24}{6} \)

This division gives us \(x = 4\). Isolating the variable makes the solution clear and straightforward. To consistently isolate variables:

• Perform inverse operations (like division if it’s multiplication, or subtraction if it’s addition).
• Whatever operation you perform on one side of the equation, do the same on the other side as well to maintain equality.

Isolating the variable helps in deriving a clear solution for \(x\) or any variable you are solving for.

Solution Verification

Verifying your solution confirms accuracy and avoids errors. With \(x = 4\), check it using the original equation:

• Substitute \(x = 4\) back into \( 24 - 5x = x \).
• Calculate the left side: \( 24 - 5 \times 4 = 24 - 20 = 4 \).
• The right side is \(x = 4\).

Both sides equal 4, confirming the solution is correct. Follow these steps for verification:

• Replace the variable with the derived value.
• Simplify both sides of the equation independently.
• Check that both sides of the equation are equal.

Solution verification is essential as it ensures the process you've done is correct and reliable.