Practice worksheet for finding linear equations from two given points.

Absolutely! We are given 10 problems where we must find the linear equation in slope-intercept form:
> y = mx + b
where:
- m is the slope (rate of change)
- b is the y-intercept (value of y when x = 0)

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✔ Step-by-step Method:

To find the equation from two points (x₁, y₁) and (x₂, y₂):

1. Calculate the slope (m):
> m = (y₂ - y₁) / (x₂ - x₁)

2. Use point-slope form to find b:
Plug one point and the slope into:
> y = mx + b → solve for b

Or use:
> y - y₁ = m(x - x₁) → then rearrange to y = mx + b

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Let’s solve each problem one by one.

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## 🔢 Problem 1: Points (1, -7) and (-5, 0)

Step 1: Find slope m

m = (0 - (-7)) / (-5 - 1) = (7) / (-6) = -7/6

Step 2: Use point (1, -7) to find b

y = mx + b
→ -7 = (-7/6)(1) + b
→ -7 = -7/6 + b
→ b = -7 + 7/6 = (-42/6 + 7/6) = -35/6

✔ Answer: y = (-7/6)x - 35/6

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## 🔢 Problem 2: Points (6, 9) and (-4, -2)

Step 1: m = (-2 - 9)/(-4 - 6) = (-11)/(-10) = 11/10

Step 2: Use (6, 9)

9 = (11/10)(6) + b
9 = 66/10 + b = 33/5 + b
b = 9 - 33/5 = 45/5 - 33/5 = 12/5

✔ Answer: y = (11/10)x + 12/5

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## 🔢 Problem 3: Points (8, -6) and (6, 8)

Step 1: m = (8 - (-6))/(6 - 8) = (14)/(-2) = -7

Step 2: Use (8, -6)

-6 = -7(8) + b
-6 = -56 + b
b = -6 + 56 = 50

✔ Answer: y = -7x + 50

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## 🔢 Problem 4: Points (4, 1) and (-6, 4)

Step 1: m = (4 - 1)/(-6 - 4) = 3/(-10) = -3/10

Step 2: Use (4, 1)

1 = (-3/10)(4) + b
1 = -12/10 + b = -6/5 + b
b = 1 + 6/5 = 11/5

✔ Answer: y = (-3/10)x + 11/5

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## 🔢 Problem 5: Points (6, -7) and (8, 5)

Step 1: m = (5 - (-7))/(8 - 6) = 12/2 = 6

Step 2: Use (6, -7)

-7 = 6(6) + b
-7 = 36 + b
b = -7 - 36 = -43

✔ Answer: y = 6x - 43

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## 🔢 Problem 6: Points (8, 7) and (-6, 4)

Step 1: m = (4 - 7)/(-6 - 8) = (-3)/(-14) = 3/14

Step 2: Use (8, 7)

7 = (3/14)(8) + b
7 = 24/14 + b = 12/7 + b
b = 7 - 12/7 = 49/7 - 12/7 = 37/7

✔ Answer: y = (3/14)x + 37/7

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## 🔢 Problem 7: Points (5, -6) and (-4, 2)

Step 1: m = (2 - (-6))/(-4 - 5) = 8/(-9) = -8/9

Step 2: Use (5, -6)

-6 = (-8/9)(5) + b
-6 = -40/9 + b
b = -6 + 40/9 = -54/9 + 40/9 = -14/9

✔ Answer: y = (-8/9)x - 14/9

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## 🔢 Problem 8: Points (-1, 6) and (8, 6)

Step 1: m = (6 - 6)/(8 - (-1)) = 0/9 = 0

This is a horizontal line!

Step 2: Since slope is 0, y = b

From either point, y = 6 → so b = 6

✔ Answer: y = 0x + 6 → y = 6

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## 🔢 Problem 9: Points (2, 7) and (0, 7)

Step 1: m = (7 - 7)/(0 - 2) = 0/-2 = 0

Again, horizontal line!

Step 2: y = 7

✔ Answer: y = 7

*(Note: Point (0,7) directly gives y-intercept b=7)*

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## 🔢 Problem 10: Points (2, -7) and (-9, 9)

Step 1: m = (9 - (-7))/(-9 - 2) = (16)/(-11) = -16/11

Step 2: Use (2, -7)

-7 = (-16/11)(2) + b
-7 = -32/11 + b
b = -7 + 32/11 = -77/11 + 32/11 = -45/11

✔ Answer: y = (-16/11)x - 45/11

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## 📝 Final Answers Summary:

1. y = (-7/6)x - 35/6
2. y = (11/10)x + 12/5
3. y = -7x + 50
4. y = (-3/10)x + 11/5
5. y = 6x - 43
6. y = (3/14)x + 37/7
7. y = (-8/9)x - 14/9
8. y = 6
9. y = 7
10. y = (-16/11)x - 45/11

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✔ All equations are now solved using the standard method: find slope → plug into y = mx + b → solve for b.

Let me know if you’d like these simplified further or written as mixed numbers or decimals!