Question

linear equations: distributive property
mining when an equation has infinite solutions
which equation is an identity?
$x - 4(x + 1) = -3(x + 1) + 1$
$2x + 3 = \frac{1}{2}(4x + 2) + 2$
$3(x - 1) = x + 2(x + 1) + 1$
$\frac{1}{3}(6x - 3) = 3(x + 1) - x - 2$

Explanation:

Step1: Simplify the first equation

Expand both sides: \(x - 4x - 4 = -3x - 3 + 1\)
Simplify: \(-3x - 4 = -3x - 2\)
Subtract \(-3x\) from both sides: \(-4 = -2\) (False, not an identity)

Step2: Simplify the second equation

Expand the right side: \(2x + 3 = 2x + 1 + 2\)
Simplify: \(2x + 3 = 2x + 3\)
Subtract \(2x\) from both sides: \(3 = 3\) (True, identity)

Step3: Simplify the third equation (assuming the typo is \(3(x - 1)\))

Expand both sides: \(3x - 3 = x + 2x + 2 + 1\)
Simplify: \(3x - 3 = 3x + 3\)
Subtract \(3x\) from both sides: \(-3 = 3\) (False, not an identity)

Step4: Simplify the fourth equation

Expand both sides: \(2x - 1 = 3x + 3 - x - 2\)
Simplify: \(2x - 1 = 2x + 1\)
Subtract \(2x\) from both sides: \(-1 = 1\) (False, not an identity)

Answer:

\(2x + 3 = \frac{1}{2}(4x + 2) + 2\) (the second equation)