Question

Find the value of $B - A$ if the graph of $Ax + By = 3$ passes through the point $(-7,2),$ and is parallel to the graph of $x + 3y = -5.$

Answer 1

-6

Explanation:

Let $A = (-3,8)$ and $B = (-5,4)$. The midpoint of $\overline{AB}$ is $\left( \frac{(-3) + (-5)}{2}, \frac{8 + 4}{2} \right) = (-4,6)$.

The slope of $\overline{AB}$ is $\frac{8 - 4}{(-3) - (-5)} = 2$, so the slope of the perpendicular bisector of $\overline{AB}$ is $-\frac{1}{2}$. Therefore, the equation of the perpendicular bisector is given by

\[y - 6 = -\frac{1}{2} (x + 4).\]Isolating $y,$ we find

\[y = -\frac{1}{2} x + 4.\]Therefore, $m+b = -\frac{1}{2} + 4 = \boxed{\frac{7}{2}}.$

Answer 2

-6

Explanation:

We know that since Ax + By = 3 passes through (-7, 2), then if we plug -7 in for x and 2 in for y, the equation is satisfied. So, let's do that:

Ax + By = 3

A * (-7) + B * 2 = 3

-7A + 2B = 3

We also know that this line is parallel to x + 3y = -5, which means their slopes are the same. Let's solve for y in the second equation:

x + 3y = -5

3y = -x - 5

y = (-1/3)x - (5/3)

So, the slope of this line is -1/3, which means the slope of Ax + By = 3 is also -1/3. Let's solve for y in the first equation:

Ax + By = 3

By = -Ax + 3

y = (-A/B)x + 3/B

This means that -A/B = -1/3. So, we have a relationship between A and B:

-A/B = -1/3

A/B = 1/3

B = 3A

Plug 3A in for B into the equation we had where -7A + 2B = 3:

-7A + 2B = 3

-7A + 2 * 3A = 3

-7A + 6A = 3

-A = 3

A = -3

Use this to solve for B:

B = 3A

B = 3 * (-3) = -9

So, B = -9 and A = -3. Then B - A is:

B - A = -9 - (-3) = -9 + 3 = -6

The answer is -6.

~ an aesthetics lover