Problem 56 - Algebra 2

Question

Solve each equation. Check your solution. $$ \frac{x}{4}=\frac{x-3}{8} $$

Step-by-Step Solution

Answer

\(x = -3\)

Step 1: Eliminate the Fractions

First, to make things easier, let's eliminate the fractions by multiplying every term by 8, which is the least common multiple of 4 and 8. Doing this gives: \(2x = x - 3\)

Step 2: Solve the equation

We now resolve the equation for \(x\) . Following the standard approach to solve this type of equations, let's isolate \(x\) on one side by subtracting \(x\) from both sides of the equation, yielding: \(x = -3\)

Step 3: Check the solution

To verify the correctness of the solution, substitute \(x = -3\) back into the original equation. If both sides equal, then the solution is correct: \(\frac{-3}{4} = \frac{-3 - 3}{8}\) simplifies to \(-0.75 = -0.75\). Thus, the solution \(x = -3\) is correct.

Key Concepts

Equation Solving

Equation solving is a fundamental skill in algebra. You start with an equation and aim to find the value of the unknown variable that makes the equation true. In this exercise, the equation is initially given as \( \frac{x}{4}=\frac{x-3}{8} \). To solve it, follow a step-by-step process:

• **Identify the terms:** Recognize both sides of the equation and the denominators involved.
• **Manipulate the equation:** Use algebraic operations like addition, subtraction, multiplication, or division to simplify the equation.
• **Isolate the variable:** Ensure the variable is by itself on one side of the equation.
• **Verify your solution:** Once the variable is isolated, check that it satisfies the original equation.

In this problem, solving the equation involves manipulating terms effectively to isolate \(x\). This requires understanding how to use inverse operations to achieve balance, ultimately solving for \(x = -3\).

Fraction Elimination

Fractions in equations can sometimes be complex to work with directly. The strategic step is to eliminate them, making the equation easier to handle. Here's how you do it:

• **Find the Least Common Multiple (LCM):** Identify the LCM of the denominators, in this case, 4 and 8, which is 8.
• **Multiply through by the LCM:** Multiply every term in the equation by the LCM. This action removes the denominators, converting fractions into whole numbers.

For the equation \( \frac{x}{4}=\frac{x-3}{8} \), multiplying every term by 8 transforms it into \(2x = x - 3\). Notice how fractions vanish and leave behind a simpler equation to solve. This transformation is not only useful for ease but also crucial for accuracy in finding the correct value for \(x\).

Verification of Solutions

Once a solution is found, verifying its correctness is important. This step ensures the solution is valid and that no mistakes were made during calculations. Verification involves checking whether the solution satisfies the original equation.Here's how to verify:

• **Substitute:** Replace the variable with the solution into the original equation.
• **Simplify both sides:** Calculate each side of the equation to confirm they are equal.

For this exercise, substituting \(x = -3\) back into the original equation \( \frac{x}{4}=\frac{x-3}{8} \), we find that both sides equal \(-0.75\). The equality of both sides confirms the solution \(x = -3\) is accurate. Remember, always ensure the solution holds true when substituted into the initial equation. This check is crucial for ensuring the integrity of your solution.