Comprehensive algebra worksheet designed to help students practice writing equations of linear functions using gradients and coordinates.

Let’s solve each part of the worksheet step by step.

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Section A: Use gradient and coordinate to find equation of line

We use the formula:
> y = mx + c, where m is gradient, and c is y-intercept.
But if we’re given a point (x, y) and gradient m, we can plug into:
> y - y₁ = m(x - x₁) → then rearrange to y = mx + c

However, notice that in Section A, all coordinates have x = 0 — meaning they are all y-intercepts! So for these, the equation is simply:
> y = mx + b, where b is the y-coordinate of the point.

So:

1. Gradient 3, (0,4) → y = 3x + 4
2. Gradient 2, (0,1) → y = 2x + 1
3. Gradient -2, (0,6) → y = -2x + 6
4. Gradient -1, (0,3) → y = -x + 3
5. Gradient 1/4, (0,3) → y = (1/4)x + 3
6. Gradient -1/4, (0,-1) → y = (-1/4)x - 1
7. Gradient 2/5, (0,-2) → y = (2/5)x - 2
8. Gradient -1/4, (0,0) → y = (-1/4)x + 0 → y = -1/4 x

Now right column:

9. Gradient -3, (0,0) → y = -3x
10. Gradient 5, (1,3) → Use y - y₁ = m(x - x₁)
y - 3 = 5(x - 1) → y - 3 = 5x - 5 → y = 5x - 2
11. Gradient 2, (-1,1) → y - 1 = 2(x + 1) → y - 1 = 2x + 2 → y = 2x + 3
12. Gradient -5, (1,2) → y - 2 = -5(x - 1) → y - 2 = -5x + 5 → y = -5x + 7
13. Gradient -2, (1,1) → y - 1 = -2(x - 1) → y - 1 = -2x + 2 → y = -2x + 3
14. Gradient -4, (4,0) → y - 0 = -4(x - 4) → y = -4x + 16
15. Gradient -1/2, (3,1) → y - 1 = (-1/2)(x - 3) → y = (-1/2)x + 3/2 + 1 → y = (-1/2)x + 5/2
16. Gradient 1/4, (3,2) → y - 2 = (1/4)(x - 3) → y = (1/4)x - 3/4 + 2 → y = (1/4)x + 5/4

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Section B: Use line equation to find gradient and y-intercept

We rearrange each equation to form y = mx + c, then m = gradient, c = y-intercept.

Left column:

1. y = 3x - 5 → Gradient = 3, y-intercept = -5
2. y = -4x + 2 → Gradient = -4, y-intercept = 2
3. y = (1/2)x → Gradient = 1/2, y-intercept = 0
4. y = x/2 - 1 → same as y = (1/2)x - 1 → Gradient = 1/2, y-intercept = -1
5. x + y = 5 → y = -x + 5 → Gradient = -1, y-intercept = 5
6. y - x = 8 → y = x + 8 → Gradient = 1, y-intercept = 8
7. x - y = 2 → -y = -x + 2 → y = x - 2 → Gradient = 1, y-intercept = -2

Right column:

8. y + 4x = 6 → y = -4x + 6 → Gradient = -4, y-intercept = 6
9. y - 2x + 9 = 0 → y = 2x - 9 → Gradient = 2, y-intercept = -9
10. 3y - x = 1 → 3y = x + 1 → y = (1/3)x + 1/3 → Gradient = 1/3, y-intercept = 1/3
11. 4x - 8y = 12 → -8y = -4x + 12 → y = (1/2)x - 3/2 → Gradient = 1/2, y-intercept = -3/2
12. y/3 + 1 = 2x → y/3 = 2x - 1 → y = 6x - 3 → Gradient = 6, y-intercept = -3
13. (y - 9x)/5 = 6 → y - 9x = 30 → y = 9x + 30 → Gradient = 9, y-intercept = 30

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Section C: Solve problems

1) Find equation with gradient 3 through (4,13)

Use y - y₁ = m(x - x₁)
y - 13 = 3(x - 4)
y - 13 = 3x - 12
y = 3x + 1

✔ Check: when x=4, y=3*4+1=13 → correct.

2) Gradient -2, y-intercept (0,-4) → so c = -4

Equation: y = -2x - 4

Now write in form ax + by + c = 0:

Bring all terms to one side:
2x + y + 4 = 0

✔ Standard form: 2x + y + 4 = 0

3) Show (2,3) lies on y = 5x - 7

Plug x=2 into equation:
y = 5*(2) - 7 = 10 - 7 = 3 → matches y=3

✔ So yes, (2,3) is on the line.

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Final Answer:

Section A Equations:
Left:
y = 3x + 4
y = 2x + 1
y = -2x + 6
y = -x + 3
y = (1/4)x + 3
y = (-1/4)x - 1
y = (2/5)x - 2
y = (-1/4)x

Right:
y = -3x
y = 5x - 2
y = 2x + 3
y = -5x + 7
y = -2x + 3
y = -4x + 16
y = (-1/2)x + 5/2
y = (1/4)x + 5/4

Section B Gradient and y-intercept:
Left:
Gradient: 3, y-int: -5
Gradient: -4, y-int: 2
Gradient: 1/2, y-int: 0
Gradient: 1/2, y-int: -1
Gradient: -1, y-int: 5
Gradient: 1, y-int: 8
Gradient: 1, y-int: -2

Right:
Gradient: -4, y-int: 6
Gradient: 2, y-int: -9
Gradient: 1/3, y-int: 1/3
Gradient: 1/2, y-int: -3/2
Gradient: 6, y-int: -3
Gradient: 9, y-int: 30

Section C Answers:
1) y = 3x + 1
2) 2x + y + 4 = 0
3) Substituting x=2 gives y=3, so point lies on line.