Question

select the two equations that represent the same line.
$y - 2 = 3(x - 1)$
$y + 4 = \frac{1}{2}(x - 2)$
$x - 2y - 10 = 0$
$y = -2x + 5$
$2x + 3y - 8 = 0$

Explanation:

Step1: Convert each equation to slope - intercept form ($y = mx + b$)

• For \(y - 2=3(x - 1)\):

Expand the right - hand side: \(y-2 = 3x-3\).
Add 2 to both sides: \(y=3x - 3 + 2=3x-1\).

• For \(y + 4=\frac{1}{2}(x - 2)\):

Expand the right - hand side: \(y + 4=\frac{1}{2}x-1\).
Subtract 4 from both sides: \(y=\frac{1}{2}x-1 - 4=\frac{1}{2}x-5\).

• For \(x - 2y-10 = 0\):

Solve for \(y\):
First, move \(x\) and \(- 10\) to the other side: \(-2y=-x + 10\).
Divide both sides by \(-2\): \(y=\frac{1}{2}x-5\).

• For \(y=-2x + 5\):

This is already in slope - intercept form with \(m=-2\) and \(b = 5\).

• For \(2x+3y - 8 = 0\):

Solve for \(y\):
First, move \(2x\) and \(-8\) to the other side: \(3y=-2x + 8\).
Divide both sides by 3: \(y=-\frac{2}{3}x+\frac{8}{3}\).

Step2: Compare the equations in slope - intercept form

We can see that the equation \(y + 4=\frac{1}{2}(x - 2)\) (which simplifies to \(y=\frac{1}{2}x-5\)) and the equation \(x - 2y-10 = 0\) (which also simplifies to \(y=\frac{1}{2}x-5\)) have the same slope (\(m=\frac{1}{2}\)) and the same y - intercept (\(b=-5\)). So these two equations represent the same line.

Answer:

\(y + 4=\frac{1}{2}(x - 2)\) and \(x - 2y-10 = 0\)