The answer is, x + 4y = 12 is y = 4x + 10.
Here is why:
To find the equation of a line that passes through the point (-3,-2) and is perpendicular to the line x+4y=12, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
First, let's rewrite the given equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
x + 4y = 12
To isolate y, subtract x from both sides:
4y = -x + 12
Divide both sides by 4 to solve for y:
y = (-1/4)x + 3
The slope of this line is -1/4.
Since we want a line that is perpendicular to this line, we need to find the negative reciprocal of the slope.
The negative reciprocal of -1/4 is 4/1 or simply 4.
Now that we have the slope of the line we want, and we know it passes through the point (-3,-2), we can use the point-slope form of a linear equation.
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope.
Substituting the values, we have:
y - (-2) = 4(x - (-3))
Simplifying:
y + 2 = 4(x + 3)
Expanding the equation:
y + 2 = 4x + 12
Subtracting 2 from both sides to isolate y:
y = 4x + 10
Therefore, the equation of the line that passes through the point (-3,-2) and is perpendicular to the line x + 4y = 12 is y = 4x + 10.