To write the standard form of the equation of a line passing through two given points, you can use the point-slope form and then rearrange the equation.
1. Find the slope of the line using the formula:
slope (m) = (y2 - y1) / (x2 - x1)
For the points M(-4, -2) and N(4, 7):
m = (7 - (-2)) / (4 - (-4))
= 9 / 8
2. Use the point-slope form of the equation of a line:
y - y1 = m (x - x1)
Select one of the given points, let's use M(-4, -2):
y - (-2) = (9/8)(x - (-4))
y + 2 = (9/8)(x + 4)
3. Simplify the equation:
Distribute 9/8 to both terms inside the parentheses:
y + 2 = (9/8)x + 9/2
4. Rearrange the equation into the standard form:
Move all the terms to the left side of the equation:
(9/8)x - y + 9/2 - 2 = 0
To eliminate the fractions, multiply the entire equation by 8:
9x - 8y + 36 - 16 = 0
Simplify the equation:
9x - 8y + 20 = 0
So, the standard form of the equation of the line passing through the points M(-4, -2) and N(4, 7) is 9x - 8y + 20 = 0.
Now let's move on to finding the slope-intercept form of the equation.
To find the slope-intercept form of the equation of a line passing through two given points, you can use the point-slope form and then rearrange the equation.
1. Find the slope of the line using the formula:
slope (m) = (y2 - y1) / (x2 - x1)
For the points P(2, -9) and Q(3, -7):
m = (-7 - (-9)) / (3 - 2)
= 2 / 1
= 2
2. Use the point-slope form of the equation of a line:
y - y1 = m (x - x1)
Select one of the given points, let's use P(2, -9):
y - (-9) = 2 (x - 2)
y + 9 = 2x - 4
3. Rearrange the equation to isolate y:
y = 2x - 4 - 9
y = 2x - 13
So, the slope-intercept form of the equation of the line passing through the points P(2, -9) and Q(3, -7) is y = 2x - 13.