Write an equation in slope-intercept form of the line satisfying the given conditions.
Through (-2, -6), perpendicular to -7x - 4y = -10
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.