How to find the equation of a line given a point and the slope ?
If given a point (x1, y1) and a slope m, then we can use the below formula to find the equation of the line.
Formula for point-slope form :
y – y1 = m(x – x1)
where,
(x1, y1) are the point on the line.
m = Slope of the line.
Find the equation of the line through :
Problem 1 :
(1,-2) having a gradient of 3
Solution :
Given, point (1, -2) and slope 3
Using point slope form formula :
Here x1 = 1, y1 = -2 and m = 3
y – y1 = m(x – x1)
(y – (-2)) = 3(x – 1)
y + 2 = 3x – 3
y = 3x – 3 – 2
y = 3x - 5
So, the required equation of a line is y = 3x - 5
Problem 2 :
(-4, -1) having a gradient of -2
Solution :
Given, point (-4, -1) and slope -2
Using point slope form formula :
Here x1 = -4, y1 = -1 and m = -2
y – y1 = m(x – x1)
(y – (-1)) = -2(x – (-4))
y + 1 = -2(x + 4)
y + 1 = -2x – 8
y = -2x – 8 – 1
y = -2x – 9
So, the required equation of a line is y = -2x – 9
Problem 3 :
(5, -2) having a gradient of -3
Solution :
Given, point (5, -2) and slope -3
Using point slope form formula :
Here x1 = 5, y1 = -2 and m = -3
y – y1 = m(x – x1)
(y – (-2)) = - 3(x – 5)
y + 2 = -3x + 15
y = -3x + 15 – 2
y = -3x + 13
So, the required equation of a line is y = -3x + 13
Problem 4 :
(5, 2) having a gradient of 1/3
Solution :
Given, point (5, 2) and slope 1/3
Here x1 = 5, y1 = 2 and m = 1/3
y – y1 = m(x – x1)
y – 2 = 1/3(x – 5)
3y – 6 = x - 5
3y = x – 5 + 6
3y = x + 1
y = x/3 + 1/3
Problem 5 :
(-2, 8) having a gradient of -1/4
Solution :
Given, point (-2, 8) and slope -1/4
Here x1 = -2, y1 = 8 and m = -1/4
y – y1 = m(x – x1)
y – 8 = -1/4(x – (-2))
y – 8 = -1/4(x + 2)
4y – 32 = -x – 2
4y = -x – 2 + 32
4y = -x + 30
y = -x/4 + 30/4
y = -x/4 + 15/2
Problem 6 :
(7, -3) having a gradient of 0
Solution :
Given, point (7, -3) and slope 0
y – y1 = m(x – x1)
(y – (-3)) = 0(x – 7)
y + 3 = 0
y = -3
Problem 7 :
(1, 6) with gradient 2/3
Solution :
Given, point (1, 6) and slope 2/3
y – y1 = m(x – x1)
y – 6 = 2/3(x – 1)
3y - 18 = 2(x – 1)
3y – 18 = 2x – 2
3y = 2x – 2 + 18
3y = 2x + 16
y = 2x/3 + 16/3
Problem 8 :
(-5, 4) with gradient 3/5
Solution :
Given, point (-5, 4) and slope 3/5
y – y1 = m(x – x1)
y – 4 = 3/5(x – (-5))
5y - 20 = 3(x + 5)
5y – 20 = 3x + 15
5y = 3x + 15 + 20
5y = 3x + 35
y = 3x/5 + 35/5
y = 3x/5 + 7
Problem 9 :
(8, 0) with gradient -1/4
Solution :
Given, point (8, 0) and slope -1/4
y – y1 = m(x – x1)
y – 0 = -1/4(x – 8)
y = -1/4(x - 8)
4y = -x + 8
y = -x/4 + 8/4
y = -x/4 + 2
Problem 10 :
(8, -2) with gradient -3/4
Solution :
Given, point (8, -2) and slope -3/4
y – y1 = m(x – x1)
(y – (-2)) = - 3/4(x – 8)
y + 2 = -3/4(x -8)
4y + 8 = -3x + 24
4y = -3x + 24 - 8
4y = -3x + 16
y = -3x/4 + 16/4
y = -3x/4 + 4
Problem 11 :
(-2, -4) with gradient 3
Solution :
Given, point (-2, -4) and slope 3
y – y1 = m(x – x1)
(y – (-4)) = 3(x – (-2))
y + 4 = 3(x + 2)
y + 4 = 3x + 6
y = 3x + 6 – 4
y = 3x + 2
Problem 12 :
(5, -1) with gradient -5
Solution :
Given, point (5, -1) and slope -5
y – y1 = m(x – x1)
(y – (-1)) = - 5(x – 5)
y + 1 = -5x + 25
y = -5x + 25 - 1
y = -5x + 24
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