Educational math worksheet focusing on identifying patterns and writing equations from input-output tables.

Let’s solve each pattern step by step. We’re looking for a rule that turns the “In (x)” number into the “Out (y)” number.

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Problem 1:

In (x): 2, 3, 4, 5
Out (y): 16, 24, 32, 40

Check differences:
- From x=2 to x=3 → y goes from 16 to 24 → +8
- x=3 to 4 → 24 to 32 → +8
- x=4 to 5 → 32 to 40 → +8

So every time x increases by 1, y increases by 8 → slope is 8.

Try equation: y = 8x + b

Plug in x=2, y=16:
16 = 8*2 + b → 16 = 16 + b → b = 0

✔ Equation: y = 8x

Check with x=3: 8*3 = 24 ✔️
x=4: 32 ✔️
x=5: 40 ✔️

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Problem 2:

In (x): 21, 18, 15, 12
Out (y): 7, 6, 5, 4

Notice: as x decreases by 3, y decreases by 1.

So when x goes down 3, y goes down 1 → so for every 1 increase in x, y increases by 1/3? That seems messy.

Wait — look at ratios:

21 ÷ 3 = 7
18 ÷ 3 = 6
15 ÷ 3 = 5
12 ÷ 3 = 4

Oh! So y = x ÷ 3

✔ Equation: y = x / 3

Check: 21/3=7 ✔️, 18/3=6 ✔️, etc.

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Problem 3:

In (x): 4, 5, 6, 7
Out (y): 10, 11, 12, 13

Each time x increases by 1, y increases by 1 → slope = 1

Try y = x + b

Use x=4, y=10: 10 = 4 + b → b = 6

✔ Equation: y = x + 6

Check: x=5 → 5+6=11 ✔️, x=6→12 ✔️, x=7→13 ✔️

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Problem 4:

In (x): 15, 10, 9, 5
Out (y): 11, 6, 5, 1

Look at differences:

From x=15 to x=10 → y from 11 to 6 → drop of 5 in x, drop of 5 in y → same change?

Wait: 15 - 4 = 11
10 - 4 = 6
9 - 4 = 5
5 - 4 = 1

Yes! Subtract 4.

✔ Equation: y = x - 4

Check all: 15-4=11 ✔️, 10-4=6 ✔️, 9-4=5 ✔️, 5-4=1 ✔️

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Problem 5:

In (x): 5, 7, 9, 11
Out (y): 15, 21, 27, 33

Check ratios:

5 × 3 = 15
7 × 3 = 21
9 × 3 = 27
11 × 3 = 33

Perfect!

✔ Equation: y = 3x

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Problem 6:

In (x): 5, 6, 8, 9
Out (y): 35, 42, 56, 63

Check:

5 × 7 = 35
6 × 7 = 42
8 × 7 = 56
9 × 7 = 63

✔ Equation: y = 7x

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Now let’s do the right side problems (smaller tables):

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Problem 7 (top left small table):

In (x): 1, 2, 4, 5
Out (y): 9, 18, 36, 45

Check:

1 × 9 = 9
2 × 9 = 18
4 × 9 = 36
5 × 9 = 45

✔ Equation: y = 9x

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Problem 8 (top middle small table):

In (x): 12, 14, 16, 17
Out (y): 4, 6, 8, 9

Check:

12 ÷ 3 = 4
14 ÷ ? → 14 ÷ 3 ≈ 4.66… no

Wait: 12 - 8 = 4
14 - 8 = 6
16 - 8 = 8
17 - 8 = 9

Yes!

✔ Equation: y = x - 8

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Problem 9 (top right small table):

In (x): 3, 5, 7, 9
Out (y): 10, 12, 14, 16

Difference: x increases by 2, y increases by 2 → so per 1 x, y increases by 1 → slope = 1

Try y = x + b

x=3, y=10 → 10 = 3 + b → b = 7

Check: x=5 → 5+7=12 ✔️, x=7→14 ✔️, x=9→16 ✔️

✔ Equation: y = x + 7

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Problem 10 (middle left small table):

In (x): 15, 10, 4, 3
Out (y): 5, 0, -6, -7

Check:

15 - 10 = 5
10 - 10 = 0
4 - 10 = -6
3 - 10 = -7

✔ Equation: y = x - 10

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Problem 11 (middle middle small table):

In (x): 2, 7, 11, 15
Out (y): 7, 21, 33, 45

Check:

2 × 3.5 = 7 → not nice

Wait: 2 → 7: difference +5
7 → 21: +14? Not consistent.

Try multiplication:

2 × 3 = 6 → close to 7
7 × 3 = 21 ✔️
11 × 3 = 33 ✔️
15 × 3 = 45 ✔️

But 2 × 3 = 6 ≠ 7 → doesn’t fit first one.

Wait — maybe it's y = 3x + 1?

Test:

x=2: 3*2 +1 = 7 ✔️
x=7: 21+1=22 ✘ → too big

Wait — what if we check differences between x and y:

For x=2, y=7 → diff = 5
x=7, y=21 → diff = 14
Not helpful.

Another idea: Look at how much y changes per x change.

From x=2 to x=7 → Δx=5, Δy=14 → slope = 14/5 = 2.8 → messy.

Wait — try dividing y by x:

7÷2 = 3.5
21÷7 = 3
33÷11 = 3
45÷15 = 3

Only first one is different. Maybe typo? Or perhaps it’s y = 3x for x≥7, but not for x=2?

Wait — let’s recheck the problem. The user image shows:

In (x): 2, 7, 11, 15
Out (y): 7, 21, 33, 45

What if it’s y = 3x + 1 for x=2? 3*2+1=7 ✔️
Then for x=7: 3*7=21 → but 3*7+1=22 ≠21 → no.

Wait — what if it’s y = 3x for all except first? But that’s not fair.

Alternative: Maybe it’s y = 3x for x>2, but let’s see if there’s another pattern.

Wait — 2 → 7: add 5
7 → 21: multiply by 3? Inconsistent.

Hold on — let’s try linear regression mentally.

Assume y = mx + b

Use two points: (7,21) and (11,33)

Slope m = (33-21)/(11-7) = 12/4 = 3

So y = 3x + b

Plug in (7,21): 21 = 21 + b → b=0

Then for x=2: y=6, but actual is 7 → off by 1.

For x=15: 3*15=45 ✔️

So only x=2 is off. Perhaps it’s a mistake? Or maybe the pattern is y = 3x for most, but let’s check if there’s another way.

Wait — what if it’s y = 3x + c, and c is not constant? No, equations should be linear.

Perhaps I misread the table. Let me double-check the original image description.

Actually, looking back — in the user’s image, for Problem 11 (which is labeled as #5 on the right page), the values are:

In (x): 2, 7, 11, 15
Out (y): 7, 21, 33, 45

But 2×3=6, not 7. Unless... wait, maybe it’s y = 3x + 1 for x=2, but then others don’t fit.

Another thought: Maybe it’s y = 3x for x≥7, and for x=2 it’s special? But that’s not a single equation.

Wait — let’s calculate the difference y - 3x:

For x=2: 7 - 6 = 1
x=7: 21 - 21 = 0
x=11: 33 - 33 = 0
x=15: 45 - 45 = 0

So only first point has +1. That suggests either a typo or perhaps the pattern starts later.

But since three out of four fit y=3x perfectly, and the first one is very close, maybe it’s intended to be y=3x, and x=2 is a distractor? Unlikely.

Wait — what if the first value is wrong? Or perhaps I need to consider non-linear? But the title is "Patterns and Equations" implying linear.

Let me try another approach: look at the ratio of y/x:

7/2 = 3.5
21/7 = 3
33/11 = 3
45/15 = 3

Average is about 3.125, not helpful.

Perhaps it’s y = 3x + k where k depends on x? No.

Wait — here’s an idea: maybe the equation is y = 3x for x > 2, but for the sake of this problem, since three points fit y=3x, and the first might be a calculation error, but let’s see the next problem.

Actually, let’s move to Problem 12 and come back.

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Problem 12 (middle right small table):

In (x): 10, 14, 16, 20
Out (y): 10, 14, 16, 20

Clearly, y = x

✔ Equation: y = x

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Back to Problem 11. Let’s list again:

x | y
2 | 7
7 | 21
11| 33
15| 45

Notice that from x=7 onwards, y=3x. For x=2, if it were 6, it would fit. But it’s 7.

What if the equation is y = 3x + 1 when x=2, but that’s not general.

Another possibility: maybe it’s y = 3x + c, and c is 1 for x=2, 0 otherwise — not possible.

Perhaps it’s a different operation. Let’s see the difference between consecutive y:

From x=2 to 7: y from 7 to 21 → +14 over +5 in x → slope 2.8

From x=7 to 11: +12 over +4 → slope 3

From x=11 to 15: +12 over +4 → slope 3

So after x=7, slope is 3. Before that, it’s different.

But for a single equation, it must be linear. So likely, the first point is meant to be x= something else, or y=6.

Given that in many such worksheets, sometimes there’s a trick, but here, let’s assume it’s y = 3x, and the first value is a typo, or perhaps I miscalculated.

Wait — what if it’s y = 3x + 1 for all? Then:

x=2: 7 ✔️
x=7: 22 ✘ (should be 21)

No.

y = 3x - 1 for x=2: 5 ✘

Another idea: perhaps it’s y = 3(x) for x>=7, and for x=2, it’s separate, but the problem asks for one equation.

Let’s calculate the average slope.

Total Δx from 2 to 15 = 13
Total Δy from 7 to 45 = 38
Slope = 38/13 ≈ 2.923 — not nice.

Perhaps it’s not linear? But the context suggests it is.

Wait — let’s look at the other problems; they are all simple linear. So probably for Problem 11, it’s intended to be y = 3x, and the first value is a mistake, or perhaps I need to see the pattern differently.

Here’s a breakthrough: what if the "In (x)" for the first row is not 2, but something else? But in the image, it’s written as 2.

Perhaps it’s y = 3x + 1 for the first, but then the equation isn't consistent.

Let’s try to fit a line using two points that work: say (7,21) and (11,33)

As before, slope = 3, intercept = 0, so y=3x.

Then for x=2, predicted y=6, actual=7 — error of 1.

For x=15, predicted 45, actual 45 — good.

So perhaps in the worksheet, it's accepted as y=3x, and the first value is to test if students notice, but typically, we go with the majority.

Maybe the pattern is y = 3x for x > 2, but for the answer, we'll put y=3x.

I recall that in some patterns, they have outliers, but here, let's check the last few problems.

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Problem 13 (bottom left small table):

In (x): 10, 8, 5, 3
Out (y): 21, 17, 11, 7

Check differences:

x=10 to 8: Δx=-2, y=21 to 17: Δy=-4 → slope = (-4)/(-2) = 2

x=8 to 5: Δx=-3, y=17 to 11: Δy=-6 → slope = 2

x=5 to 3: Δx=-2, y=11 to 7: Δy=-4 → slope = 2

So slope = 2

Equation: y = 2x + b

Use x=10, y=21: 21 = 2*10 + b → 21=20+b → b=1

Check x=8: 2*8 +1 = 17 ✔️
x=5: 10+1=11 ✔️
x=3: 6+1=7 ✔️

✔ Equation: y = 2x + 1

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Problem 14 (bottom middle small table):

In (x): 20, 16, 12, 8
Out (y): 18, 14, 10, 6

Differences:

x↓4, y↓4 → slope = 1

y = x + b

x=20, y=18: 18 = 20 + b → b = -2

Check x=16: 16-2=14 ✔️
x=12: 10 ✔️
x=8: 6 ✔️

✔ Equation: y = x - 2

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Problem 15 (bottom right small table):

In (x): 10, 12, 15, 17
Out (y): 13, 15, 18, 20

Differences:

x=10 to 12: +2, y=13 to 15: +2 → slope=1

x=12 to 15: +3, y=15 to 18: +3 → slope=1

x=15 to 17: +2, y=18 to 20: +2 → slope=1

So y = x + b

x=10, y=13: 13 = 10 + b → b=3

Check x=12: 12+3=15 ✔️
x=15: 18 ✔️
x=17: 20 ✔️

✔ Equation: y = x + 3

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Now back to Problem 11. Given that all other problems have clean linear equations, and for Problem 11, three out of four points satisfy y=3x, and the first point (2,7) is very close (off by 1), it's likely that the intended equation is y=3x, and perhaps there's a typo in the worksheet, or in some contexts, they accept it.

But let's double-check the values. Is it possible that for x=2, y should be 6? But in the image, it's written as 7.

Another possibility: maybe the equation is y = 3x + 1 for x=2, but that's not general.

Wait — what if it's y = 3x for x even, but 7 is odd — no.

Perhaps it's y = 3x + c where c is based on parity, but that's complicated.

Let's calculate the mean of y/x: (3.5 + 3 + 3 + 3)/4 = 12.5/4 = 3.125 — not helpful.

I think for the purpose of this exercise, since the majority fit y=3x, and it's a common pattern, I'll go with y = 3x for Problem 11, noting that the first point might be an exception or error.

But to be precise, let's see if there's a linear equation that fits all.

Assume y = mx + b

Use points (2,7) and (7,21):

m = (21-7)/(7-2) = 14/5 = 2.8

Then y - 7 = 2.8(x - 2)

y = 2.8x - 5.6 + 7 = 2.8x + 1.4

Check for x=11: 2.8*11 +1.4 = 30.8 +1.4 = 32.2 ≠ 33

Close but not exact.

Use (7,21) and (11,33): m=3, b=0, as before.

Then for x=2, y=6, but actual 7.

The difference is small, but in math problems, we expect exact fit.

Perhaps the first "In (x)" is 3, not 2? If x=3, y=9, but it's 7.

Another idea: maybe it's y = 3x + 1 for the first, but then the equation isn't consistent.

Let's look at the sequence of y: 7,21,33,45

Differences: 14,12,12 — not arithmetic.

Second differences: -2,0 — not constant.

So not quadratic.

Perhaps it's piecewise, but unlikely.

I recall that in some worksheets, they have patterns like y = 3x for x>2, but for the answer, we'll put y=3x, as it's the best fit.

To confirm, let's see the answer choices or typical patterns.

Since this is for a student, and the other problems are straightforward, I think for Problem 11, the intended answer is y = 3x, and the first value might be a distractor or error.

So I'll proceed with that.

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Summary of all equations:

Left page:
1. y = 8x
2. y = x / 3
3. y = x + 6
4. y = x - 4
5. y = 3x
6. y = 7x

Right page:
7. y = 9x
8. y = x - 8
9. y = x + 7
10. y = x - 10
11. y = 3x [with note that first point is slightly off, but fits others]
12. y = x
13. y = 2x + 1
14. y = x - 2
15. y = x + 3

For Problem 11, upon second thought, let's calculate exactly.

If we use the first and last point: (2,7) and (15,45)

m = (45-7)/(15-2) = 38/13 ≈ 2.923

b = y - mx = 7 - (38/13)*2 = 7 - 76/13 = (91 - 76)/13 = 15/13

Then for x=7: y = (38/13)*7 + 15/13 = (266 + 15)/13 = 281/13 ≈ 21.615 ≠ 21

Not good.

Use (7,21) and (15,45): m= (45-21)/(15-7) = 24/8 = 3, b=0, as before.

Then for x=2, y=6, but given 7.

Perhaps the equation is y = 3x + 1 when x=2, but that's not a single equation.

I think there might be a typo in the problem, and it's meant to be x=3 for the first row, but it's written as 2.

In many online sources, similar problems have y=3x for such sequences.

So I'll stick with y = 3x for Problem 11.

Final Answer:

1. y = 8x
2. y = x / 3
3. y = x + 6
4. y = x - 4
5. y = 3x
6. y = 7x
7. y = 9x
8. y = x - 8
9. y = x + 7
10. y = x - 10
11. y = 3x
12. y = x
13. y = 2x + 1
14. y = x - 2
15. y = x + 3