Write the slope-intercept equation for the line that passes through (-14, -3) and is perpendicular to -7x – 4y = -4

Question

Write the slope-intercept equation for the line that passes through (-14, -3) and is perpendicular to -7x – 4y = -4

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Answer 1

The first step to answering this question is to find the slope of the first line. To do that, we need to put it in slope-intercept form.
-7x - 4y = -4  =>  -4y = 7x - 4 => y = (-7/4)x + 1.
So the slope of this line is -7/4. Perpendicular lines have a negative reciprocal slope, so the slope of the perpendicular line is 4/7. Using that and the point (-14, -3) we can make an equation for the line in point-slope form: y + 3 = (4/7)(x + 14) . We can then convert this into slope intercept form.

y + 3 = (4/7)(x + 14) => y + 3 = (4/7)x + 8 => y = (4/7)x + 5.

So your final answer is y = 4/7 x + 5

Answer 2

Lines can be defined in three ways:

1) the standard form Ax + By = C

2) the slope-intercept form Y = mX + b where m is the slope of the line and b is the Y intercept

3) the point slope formula with uses a point and the slope of the line in the form Y-Y1 = m (X-X1)

The point you state is (-14, -3)  so y= -3 and X = -14

"Perpendicular lines have slopes that are negative reciprocals of each other." What does that mean? It means that a pair of lines that intersect at right angles will have slopes that go in opposite directions and in opposite ways. Where line one has a slope of m, which can also be written m/1,  the perpendicular line has a slope of -1/m.

The line you state is -7X - 4Y = - 4

Let's rewrite this as - 4Y = -4 + -7x  (we added -7x to each side canceling it out on the left)

Not being a fan of negative numbers I would multiply both sides by (-1)

(-1) (-4Y) = -1 (-4 -7X) ----remember that + (-7) = -7

That becomes 4Y = 4 - (-7X)   which equals 4Y = 4+7X or    4Y = 7X + 4

Now divide each side by 4, canceling it out on the left.

Y = (7X + 4) / 4

Now distribute the dividend to get Y =  7/4 X + 4/4 or    Y = 7/4X + 1

That is a form you recognize as the Y intercept formula  Y = mX + b  where m = the slope

The perpaendicular line's slope will then be the negative reciprocol of 7/4   or   - 4/7 (flip it and change the sign)

The point can now be used with this slope to write the equation of the line in the Point -Slope formula required to be compliant with the question.

Y - (-3) = - 4/7 {(X - (-14)}

This becomes Y + 3 = - 4/7 X - 56/7   but on standardized exams this usually is not going to be the final answer. 

Subtract 3 from both sides gives you Y = - 4/7 X -56/7 - 3/1

But you must find a common denominator to subtract fractions  so it is - 4/7 X - 56/7 -21/7

or Y = - 4/7 X - 35/7    or Y = - 4/7 X -5

Whew!  That's a long answer and there are many short cuts that would have made this faster to explain but when you are learning algebra short cuts are a roadblock to becoming a math-master!  Honor yourself and take it step by step. This is your time to become gap-free in math and anytime I can help, give me a call and schedule a session of academic coaching.

Dr. Carranza