Question

Solve the equation by cross multiplying. Check your solution(s). $$\frac{8}{3 x-2}=\frac{2}{x-1}$$

Short answer

The solution to the equation is \(x = 2\).

Step-by-step solution

Cross Multiply

Multiply \(8\) by \(x - 1\) and \(2\) by \(3x - 2\). This gives the equation \(8(x - 1) = 2(3x - 2)\). Expand both sides to get \(8x - 8 = 6x - 4\).

Simplify the equation to standard form

Re-arrange the equation from step 1 by subtracting \(6x\) from both sides and adding \(8\) to both sides. This gives the equation \(8x - 6x = -4 + 8\), subsequently simplifying to \(2x = 4\)

Solve the simplified equation

Solve the equation for \(x\) by dividing both sides by \(2\). This gives \(x = 4 / 2\) which simplifies to \(x = 2\)

Check your solution

Check the solution by substituting \(x = 2\)into the original equation to confirm if the equation holds true. \(LHS = 8 / (3 * 2 - 2) = 8 / 4 = 2\) and \(RHS = 2 / (2 - 1) = 2 / 1 = 2\). Since \(LHS = RHS\), the solution is valid.

Key concepts

Cross Multiplication

Cross multiplication is a technique used to solve proportions or equations where each side is a fraction. In this method, the denominators are effectively "eliminated" by multiplying each numerator by the opposite denominator.

Given the example equation: \( \frac{8}{3x-2} = \frac{2}{x-1} \), the goal is to find the value of \(x\). By cross multiplying, we form a new equation without fractions:

• Multiply the numerator on the left (\(8\)) by the denominator on the right (\(x-1\)). This results in \(8(x-1)\).
• Multiply the numerator on the right (\(2\)) by the denominator on the left (\(3x-2\)). This results in \(2(3x-2)\).

Now we have the equation \(8(x-1) = 2(3x-2)\). The absence of fractions makes it easier to solve the equation step by step.

Equation Verification

Equation verification is a crucial step when solving equations. It ensures that the calculated solution indeed satisfies the original equation. It's like a double-check to confirm the correctness of the result.

After solving the equation \( \frac{8}{3x-2} = \frac{2}{x-1} \) and finding \( x = 2 \), we substitute \(x\) back into the original equation for verification:

• Calculate the left-hand side (LHS): \( \frac{8}{3 \times 2 - 2} = \frac{8}{4} = 2 \).
• Calculate the right-hand side (RHS): \( \frac{2}{2 - 1} = \frac{2}{1} = 2 \).

Both sides result in \(2\), verifying that \(x = 2\) is indeed the correct solution. Verification gives confidence in the solution's accuracy and is an essential part of algebraic problem-solving.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to isolate the variable of interest. It's a step-by-step approach that translates complex expressions into simpler, more approachable forms.

In our exercise, after cross multiplying, we had the equation \(8(x-1) = 2(3x-2)\). First, expand both sides:

• Left side: \(8(x-1) = 8x - 8\)
• Right side: \(2(3x-2) = 6x - 4\)

These expansions simplify to an equation \(8x - 8 = 6x - 4\). The next step is rearranging the terms to isolate \(x\):

• Subtract \(6x\) from both sides: \(8x - 6x = -4 + 8\)
• Simplify to \(2x = 4\).

Finally, divide by \(2\) to find \(x = 2\). Each algebraic manipulation brings us closer to solving for \(x\), demonstrating the structured logic inherent in algebra.