Write the equation of the line that satisfies the given conditions. Express the final equation in standard form.
Contains the point (-7, 2) and is perpendicular to the line 4x − y = 4
line given is
4x − y = 4
rearranged that is
y = 4 x - 4 so slope = 4
new slope = -1/old slope
so
m = -1/4
y = -x/4 + b
to find b put the point in
2 = -(1/4)(-7) + b
solve for b
To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line. The given line is in the form Ax + By = C, where A, B, and C are constants. We can rewrite the equation to slope-intercept form (y = mx + b) to determine the slope.
4x - y = 4
Rearrange the equation to isolate y:
-y = -4x + 4
Divide through by -1 to make y positive:
y = 4x - 4
From this equation, we can see that the slope of the given line is 4 (the coefficient of x).
Since the line we want to find is perpendicular, its slope will be the negative reciprocal of 4. Therefore, the slope of the new line will be -1/4.
Now we have the slope (-1/4) and a point (-7, 2) that the line must pass through. We can use the point-slope form of a line to find the equation:
y - y1 = m(x - x1)
Substitute the values into the equation:
y - 2 = (-1/4)(x - (-7))
Simplify:
y - 2 = (-1/4)(x + 7)
Distribute the -1/4:
y - 2 = (-1/4)x - 7/4
Move the constant term (-2) to the other side:
y = (-1/4)x - 7/4 + 2
Combine the constants:
y = (-1/4)x - 7/4 + 8/4
y = (-1/4)x + 1/4
To express the equation in standard form, we need to multiply through by 4 to eliminate the fraction:
4y = -x + 1
Rearrange the equation:
x + 4y = 1
Therefore, the equation of the line that contains the point (-7, 2) and is perpendicular to the line 4x - y = 4 is x + 4y = 1 in standard form.