Question

Find \(x\): \(4(x - 2) = 3(x + 5)\).

• A. \(23\)
• B. \(17\)
• C. \(13\)
• D. \(7\)

Hint:

First expand both brackets using the distributive property, then bring all x-terms to one side and the constants to the other side before dividing to isolate x. Keep track of your signs carefully when moving terms across the equals sign.

Answer 1

The Correct Option is A

Concept: To solve linear equations, expand the brackets and collect like terms on one side. A linear equation generally has the form: \[ ax + b = cx + d \] The goal is to isolate the variable \(x\).

Step 1: Expand both sides of the equation. \[ 4(x-2) = 3(x+5) \] \[ 4x - 8 = 3x + 15 \]

Step 2: Move the variable terms to one side. \[ 4x - 3x - 8 = 15 \] \[ x - 8 = 15 \]

Step 3: Isolate \(x\). \[ x = 15 + 8 \] \[ x = 23 \] Thus, the value of \(x\) is: \[ x = 23 \]

Answer 2

Concept:

• Any equation of the form $a(x - p) = b(x + q)$ can be solved with one general formula instead of expanding fresh each time: $x = \frac{bq + ap}{a - b}$.
• This formula comes directly from expanding once and rearranging, and it can be reused for any equation written in this shape.

Step 1: Match the given equation to the general shape $a(x-p) = b(x+q)$.
$4(x-2) = 3(x+5)$ gives $a = 4$, $p = 2$, $b = 3$, $q = 5$.

Step 2: Derive the formula once, in general.
$ax - ap = bx + bq$
$ax - bx = bq + ap$
$x(a - b) = bq + ap$
$x = \frac{bq + ap}{a - b}$

Step 3: Substitute the matched values into the formula.
$x = \frac{(3)(5) + (4)(2)}{4 - 3} = \frac{15 + 8}{1}$

Final Answer: $x = 23$