Question

Which of the choices shows how to rewrite the equation \(2 y-3(y-2 x)=8(x-1)+3\) so that \(y\) is a function of \(x ?\) (A) \(x=\frac{5-y}{2}\) (B) \(x=\frac{-y+5}{10}\) (C) \(y=-14 x+5\) (D) \(y=-2 x+5\)

Short answer

The equation can be rewritten as \(y = -2x + 5\), so the choice (D) is the correct answer.

Step-by-step solution

Expand the expressions

First, apply the distributive property, \( a(b + c) = ab + ac \) and \( a(b - c) = ab - ac \) to simplify the equation. Expand \(3(y - 2x)\) and \(8(x - 1)\). This turns the equation into: \(2y - 3y + 6x = 8x - 8 + 3\).

Simplify the equation

After expanding, simplify the obtained equation by combining like terms on both sides. On the left, combine \(2y\) and \(-3y\) to get \(-y\), and keep the \(6x\). The right side simplifies to \(8x - 5\). The simplified equation now is: \(-y +6x = 8x -5\).

Isolate y on one side

The aim is to express y as a function of x. Thus, isolate y on the left side of the equation. Subtract \(6x\) from both sides of the equation to move \(x\) related terms to one side. This would give: \(-y = 2x - 5\).

Finalise the expression for y

Reverse the sign by multiplying \(-1\) to both sides of the equation, therefore obtaining the final form: \(y = -2x + 5\).