Write an equation in​ slope-intercept form of the line satisfying the given conditions.

Question

Through ​(-2​, -6​), perpendicular to -7x - 4y = -10

Answer 1

To find the equation of a line that is perpendicular to another line, we need to find the negative reciprocal of the slope of the given line.

First, let's rearrange the given equation -7x - 4y = -10 into slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept.

-7x - 4y = -10

Subtract -7x from both sides:

-4y = 7x - 10

Divide both sides by -4:

y = (-7/4)x + 5/2

So, the slope of the given line is -7/4.

The negative reciprocal of -7/4 is 4/7 (flipping the fraction sign and taking the reciprocal).

Now, we know that the slope of the line we want is 4/7, and it passes through (-2, -6).

Let's use the point-slope form of a line to find the equation:

y - y₁ = m(x - x₁)

where (x₁, y₁) is the given point (-2, -6) and m is the slope.

Plugging in the values:

y - (-6) = (4/7)(x - (-2))

Simplifying:

y + 6 = (4/7)(x + 2)

Distributing the 4/7:

y + 6 = (4/7)x + 8/7

Subtracting 6 from both sides:

y = (4/7)x + 8/7 - 6

Simplifying:

y = (4/7)x + 8/7 - 42/7

Combining the fractions:

y = (4/7)x - 34/7

So, the equation of the line that is perpendicular to -7x - 4y = -10 and passes through (-2, -6) is y = (4/7)x - 34/7.