1. To find the equation of a line that is perpendicular to a given line, you need to determine the slope of the given line and then take the negative reciprocal of that slope. After that, you can use the point-slope form or convert it to the slope-intercept form.
Given line: x = -(7/8)y + 3
To find the slope of this line, you can rewrite it in slope-intercept form (y = mx + b). Rewrite the given line:
x = -(7/8)y + 3
Multiply both sides by 8 to get rid of the fraction:
8x = -7y + 24
Rearrange terms:
7y = -8x + 24
Divide both sides by 7:
y = (-8/7)x + (24/7)
The slope of the given line is -8/7. The slope of a line perpendicular to this one will be the negative reciprocal, which is 7/8. Now use the point-slope form to find the equation:
Point: P(3, -4)
Slope: 7/8
y - y1 = m(x - x1)
y - (-4) = (7/8)(x - 3)
Simplify:
y + 4 = (7/8)(x - 3)
Lastly, convert it to slope-intercept form (y = mx + b):
y + 4 = (7/8)x - (7/8)3
y + 4 = (7/8)x - 21/8
y = (7/8)x - 21/8 - 32/8
y = (7/8)x - 53/8
Therefore, the equation of the line that passes through point P(3, -4) and is perpendicular to the line x = -(7/8)y + 3 is y = (7/8)x - 53/8 in slope-intercept form.
2. To find the equation of a line passing through two given points, you can apply the formula for slope (m) and then use the point-slope form or convert it to the slope-intercept form.
Given points: P(6, 0), Q(7, -5)
To find the slope (m), use the formula:
m = (y2 - y1) / (x2 - x1)
Substituting the values from the points:
m = (-5 - 0) / (7 - 6)
m = -5
Now use the slope-intercept form to find the equation:
Point: P(6, 0)
Slope: -5
y - y1 = m(x - x1)
y - 0 = -5(x - 6)
Simplify:
y = -5(x - 6)
y = -5x + 30
Therefore, the equation of the line passing through points P(6, 0) and Q(7, -5) is y = -5x + 30 in slope-intercept form.
3. When the slope (m) is undefined, it means the line is a vertical line. A vertical line has the equation of the form x = c, where c is a constant. To find the equation of the line passing through P(-7, -9), you can use the x-coordinate of the point as the constant in the equation:
x = -7
Therefore, the equation of the line with an undefined slope passing through P(-7, -9) is x = -7, written in standard form.