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Answers ยท Topic 5 of 6

โœ… Answer Key: Writing Linear Equations

โš ๏ธ Top Mistakes on This Topic

  • Finding b incorrectly: After finding m, substitute ONE point into y = mx + b and solve for b โ€” don't guess.
  • Forgetting to verify: Always plug your answer back into the other point to check it works for BOTH points.
  • Table questions โ€” confusing m with b: m = rate of change (ฮ”yรทฮ”x). b = y-value when x=0. If x=0 isn't in the table, substitute a point to find b.
  • Sign errors with negative slopes: If line goes DOWN from left to right, slope is negative. Double-check your subtraction order.
Q1 Easy
slope=3, through (0,2)
1
(0,2) means x=0, so this is the y-intercept โ†’ b=2
2
y = 3x + 2
y = 3x + 2
Q2 Easy
m=โˆ’2, b=5
1
Substitute directly: y = โˆ’2x + 5
y = โˆ’2x + 5
Q3 Easy
m=1/3, through (0,โˆ’1)
1
b = โˆ’1 (x=0 given). y = (1/3)x โˆ’ 1
y = (1/3)x โˆ’ 1
Q4 Medium
Through (2,7) and (4,11)
1
m = (11โˆ’7)รท(4โˆ’2) = 4รท2 = 2
2
Use (2,7): 7 = 2(2) + b โ†’ 7 = 4 + b โ†’ b = 3
3
Verify with (4,11): y = 2(4)+3 = 11 โœ“
y = 2x + 3
Q5 Medium
Through (โˆ’1,4) and (3,โˆ’4)
1
m = (โˆ’4โˆ’4)รท(3โˆ’(โˆ’1)) = โˆ’8รท4 = โˆ’2
2
Use (โˆ’1,4): 4 = โˆ’2(โˆ’1) + b โ†’ 4 = 2 + b โ†’ b = 2
3
Verify with (3,โˆ’4): y = โˆ’2(3)+2 = โˆ’4 โœ“
y = โˆ’2x + 2
Q6 Medium
Len's balloon table: x: 2,4,6,8 / y: 3,7,11,15
1
m = (7โˆ’3)รท(4โˆ’2) = 4รท2 = 2
2
x=0 not in table. Use (2,3): 3 = 2(2) + b โ†’ 3 = 4 + b โ†’ b = โˆ’1
3
Verify with (6,11): y = 2(6)โˆ’1 = 11 โœ“
y = 2x โˆ’ 1
Q7 Medium
Miku's cat table: x: โˆ’2,0,2,4 / y: 8,2,โˆ’4,โˆ’10
1
m = (2โˆ’8)รท(0โˆ’(โˆ’2)) = โˆ’6รท2 = โˆ’3
2
When x=0, y=2 โ†’ b=2
3
Verify with (4,โˆ’10): y = โˆ’3(4)+2 = โˆ’10 โœ“
y = โˆ’3x + 2
Q8 Medium
Through (โˆ’4,5) and (2,โˆ’4)
1
m = (โˆ’4โˆ’5)รท(2โˆ’(โˆ’4)) = โˆ’9รท6 = โˆ’3/2
2
Use (2,โˆ’4): โˆ’4 = (โˆ’3/2)(2) + b โ†’ โˆ’4 = โˆ’3 + b โ†’ b = โˆ’1
3
Verify with (โˆ’4,5): y = (โˆ’3/2)(โˆ’4)โˆ’1 = 6โˆ’1 = 5 โœ“
y = โˆ’(3/2)x โˆ’ 1
Q9 Hard
Through (6,1) and (โˆ’3,7)
1
m = (7โˆ’1)รท(โˆ’3โˆ’6) = 6รท(โˆ’9) = โˆ’2/3
2
Use (6,1): 1 = (โˆ’2/3)(6) + b โ†’ 1 = โˆ’4 + b โ†’ b = 5
3
Verify with (โˆ’3,7): y = (โˆ’2/3)(โˆ’3)+5 = 2+5 = 7 โœ“
y = โˆ’(2/3)x + 5
Q10 Hard
Table: x: โˆ’6,โˆ’2,2,6 / y: โˆ’9,โˆ’1,7,15 โ€” write equation, predict x=10
1
m = (โˆ’1โˆ’(โˆ’9))รท(โˆ’2โˆ’(โˆ’6)) = 8รท4 = 2
2
x=0 not in table. Use (โˆ’2,โˆ’1): โˆ’1 = 2(โˆ’2)+b โ†’ โˆ’1 = โˆ’4+b โ†’ b = 3
3
Verify with (6,15): y = 2(6)+3 = 15 โœ“
4
Predict x=10: y = 2(10)+3 = 23
y = 2x + 3  |  When x=10: y = 23

๐Ÿค– AI Prompts for Topic 5

If you don't know how to find b when x=0 isn't in the table:"How do I find the y-intercept b in y=mx+b when the table of values doesn't include x=0? Walk me through substituting a known point."
If you keep making errors with two-point problems:"I'm writing linear equations from two points. Can you show me the full process slowly: label points, calculate slope, substitute to find b, then verify with the second point?"
For fraction slopes:"I got a fraction slope like โˆ’3/2 when writing an equation. How do I use it correctly when substituting into y = mx + b to find b?"

๐Ÿ“บ Learn More

  • Khan Academy: Search "writing slope-intercept equations" and "linear equations from two points"
  • YouTube: Search "write equation of a line given two points Algebra 1"
  • Khan Academy: Search "linear equations from tables" for table-based practice