Let’s solve each problem step by step.
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Problem 4: Find the equation of the line that passes through the given point and has the given gradient.
We use the
point-slope form of a line:
>
y - y₁ = m(x - x₁)
where (x₁, y₁) is the point, and m is the gradient.
Then we can rearrange to get it in slope-intercept form:
y = mx + b
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a) Point (1, 4), gradient = 3
Start with:
y - 4 = 3(x - 1)
Expand:
y - 4 = 3x - 3
Add 4 to both sides:
y = 3x + 1
✔ Final answer for 4a:
y = 3x + 1
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b) Point (-3, 7), gradient = -2
Start with:
y - 7 = -2(x - (-3)) → y - 7 = -2(x + 3)
Expand:
y - 7 = -2x - 6
Add 7 to both sides:
y = -2x + 1
✔ Final answer for 4b:
y = -2x + 1
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c) Point (6, 2), gradient = 6
Start with:
y - 2 = 6(x - 6)
Expand:
y - 2 = 6x - 36
Add 2 to both sides:
y = 6x - 34
✔ Final answer for 4c:
y = 6x - 34
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d) Point (3, 5), gradient = -4
Start with:
y - 5 = -4(x - 3)
Expand:
y - 5 = -4x + 12
Add 5 to both sides:
y = -4x + 17
✔ Final answer for 4d:
y = -4x + 17
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Problem 5: Find the equation of the line that passes through two points.
First, find the gradient using:
>
m = (y₂ - y₁) / (x₂ - x₁)
Then use point-slope form with one of the points.
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a) Points (3, 7) and (2, -2)
Find gradient:
m = (-2 - 7) / (2 - 3) = (-9) / (-1) = 9
Use point (3, 7):
y - 7 = 9(x - 3)
Expand:
y - 7 = 9x - 27
Add 7:
y = 9x - 20
✔ Final answer for 5a:
y = 9x - 20
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b) Points (2, -3) and (6, 1)
Gradient:
m = (1 - (-3)) / (6 - 2) = (4) / (4) = 1
Use point (2, -3):
y - (-3) = 1(x - 2) → y + 3 = x - 2
Subtract 3:
y = x - 5
✔ Final answer for 5b:
y = x - 5
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c) Points (6, 7) and (5, -1)
Gradient:
m = (-1 - 7) / (5 - 6) = (-8) / (-1) = 8
Use point (6, 7):
y - 7 = 8(x - 6)
Expand:
y - 7 = 8x - 48
Add 7:
y = 8x - 41
✔ Final answer for 5c:
y = 8x - 41
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d) Points (1, 0) and (2, -3)
Gradient:
m = (-3 - 0) / (2 - 1) = (-3)/1 = -3
Use point (1, 0):
y - 0 = -3(x - 1) → y = -3x + 3
✔ Final answer for 5d:
y = -3x + 3
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Final Answer:
4.
a) y = 3x + 1
b) y = -2x + 1
c) y = 6x - 34
d) y = -4x + 17
5.
a) y = 9x - 20
b) y = x - 5
c) y = 8x - 41
d) y = -3x + 3
Parent Tip: Review the logic above to help your child master the concept of equations of a line worksheet.
