Question

solving linear equations: distributive property
which equation has no solution?
$4(x + 3) + 2x = 6(x + 2)$
$5 + 2(3 + 2x) = x + 3(x + 1)$
$5(x + 3) + x = 4(x + 3) + 3$
$4 + 6(2 + x) = 2(3x + 8)$

Explanation:

Step1: Analyze first equation

Expand both sides: \(4(x + 3)+2x = 4x + 12 + 2x = 6x + 12\); \(6(x + 2)=6x + 12\). Both sides are equal, so infinite solutions.

Step2: Analyze second equation

Expand: \(5 + 2(3 + 2x)=5 + 6 + 4x = 11 + 4x\); \(x + 3(x + 1)=x + 3x + 3 = 4x + 3\). Subtract \(4x\): \(11 = 3\), which is false. No solution.

Step3: Analyze third equation

Expand: \(5(x + 3)+x = 5x + 15 + x = 6x + 15\); \(4(x + 3)+3 = 4x + 12 + 3 = 4x + 15\). Subtract \(4x + 15\): \(2x = 0\), \(x = 0\). Has solution.

Step4: Analyze fourth equation

Expand: \(4 + 6(2 + x)=4 + 12 + 6x = 16 + 6x\); \(2(3x + 8)=6x + 16\). Both sides equal, infinite solutions.

Answer:

\(5 + 2(3 + 2x) = x + 3(x + 1)\)