Algebra - Mathematical Musings - Free Printable

Let’s solve each row one by one. We’ll look at what’s given and figure out the value of y or complete the equation based on the diagram or operation shown.

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Row 1:
Given:
> 12 = y/4

To find y, multiply both sides by 4:
→ y = 12 × 4 = 48

✔ So, y = 48

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Row 2:
Diagram shows:
- Top bar: 8 (blue)
- Bottom bars: pink “y” + yellow “5” → total should equal top bar?
So: y + 5 = 8
→ y = 8 - 5 = 3

✔ So, y = 3

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Row 3:
Operation:
> y → 12
> +14

This means: start with y, add 14 to get 12? That doesn’t make sense unless it’s reversed.

Wait — maybe it’s saying: if you take y and do something, you get 12 after adding 14? Or perhaps it’s a flow: input y, then +14 gives output 12?

That would mean:
y + 14 = 12
→ y = 12 - 14 = -2

But negative numbers might be too advanced? Let’s check context.

Alternatively, maybe it’s: starting from y, you go to 12 by adding 14? Then y must be less than 12.

Yes: y + 14 = 12 → y = -2

But let’s hold that thought — maybe in this worksheet, they’re using arrows to show operations applied to y to reach a result.

Looking ahead, Row 6 says:
> y → 40
> x4 → so y × 4 = 40 → y=10

Similarly, Row 9:
> y → ÷4 → y=3 → so y÷4=3 → y=12

So pattern: arrow means “apply operation to y to get the number on right”

So for Row 3:
> y → 12
> +14 → means: y + 14 = 12 → y = 12 - 14 = -2

We’ll keep it as -2 unless told otherwise.

✔ So, y = -2

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Row 4:
> y → 12
> –14 → means: y - 14 = 12 → y = 12 + 14 = 26

✔ So, y = 26

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Row 5:
Diagram:
- Top bar: y (blue)
- Bottom: five pink boxes, each labeled “2” → total = 2+2+2+2+2 = 10

So: y = 10

✔ So, y = 10

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Row 6:
> y → 40
> x4 → means: y × 4 = 40 → y = 40 ÷ 4 = 10

✔ So, y = 10

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Row 7:
Diagram:
- Top bar: 36 (blue)
- Bottom: three pink boxes, each “y” → so 3y = 36 → y = 36 ÷ 3 = 12

✔ So, y = 12

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Row 8:
> y → x5 → y = 50
Means: y × 5 = 50 → y = 50 ÷ 5 = 10

✔ So, y = 10

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Row 9:
> y → ÷4 → y = 3
Means: y ÷ 4 = 3 → y = 3 × 4 = 12

✔ So, y = 12

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Row 10:
Only says: y = 10 → no operation or diagram. Just state it.

✔ So, y = 10

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Now let’s compile all answers per row:

Row 1: y = 48
Row 2: y = 3
Row 3: y = -2
Row 4: y = 26
Row 5: y = 10
Row 6: y = 10
Row 7: y = 12
Row 8: y = 10
Row 9: y = 12
Row 10: y = 10

Wait — but looking back at Row 3 and Row 4, maybe the arrow direction is different?

In Row 6: “y → 40, x4” → we interpreted as y×4=40 → y=10 ✔️

In Row 9: “y → ÷4, y=3” → y÷4=3 → y=12 ✔️

So same logic applies to Rows 3 and 4.

Row 3: “y → 12, +14” → y + 14 = 12 → y = -2

Row 4: “y → 12, –14” → y - 14 = 12 → y = 26

These are mathematically correct, even if negative.

But let’s double-check if maybe the arrow means “to get y, do this to the number”.

For example, in Row 3: maybe “start with 12, subtract 14 to get y”? But the arrow points from y to 12.

The notation is:
“y → [result]” with operation below.

So operation is applied to y to get the result.

Thus, our interpretation holds.

However, in some curricula, they might write it as “input → operation → output”, so y is input, operation is done, output is 12.

So yes, y + 14 = 12 → y = -2

Same for others.

Final Answers per row:

1. y = 48
2. y = 3
3. y = -2
4. y = 26
5. y = 10
6. y = 10
7. y = 12
8. y = 10
9. y = 12
10. y = 10

But wait — Row 10 just says “y = 10” — no question mark, so probably just confirming.

Also, note that in Row 5, diagram clearly shows y = sum of five 2s → 10.

All consistent.

One more check: Row 2 diagram — blue bar is 8, bottom is y and 5 side by side → so y + 5 = 8 → y=3 ✔️

Row 7: top is 36, bottom is three y’s → 3y=36 → y=12 ✔️

Everything checks out.

Final Answer:
Row 1: y = 48
Row 2: y = 3
Row 3: y = -2
Row 4: y = 26
Row 5: y = 10
Row 6: y = 10
Row 7: y = 12
Row 8: y = 10
Row 9: y = 12
Row 10: y = 10

But since the user didn’t ask to list them all, and the image seems to have blanks to fill, perhaps we need to provide the missing values in order.

Looking at the grid, each row has one unknown — either y or an operation result.

Actually, re-examining the original image description:

It's a table with 10 rows. Each row has some info, and we need to find y or complete the relation.

From left to right, column 1 sometimes has equation, column 2 has diagram, column 3 has operation flow, column 4 has final y value.

But in many rows, only one part is given, and we deduce y.

Since the instruction is to "solve the problem", and the problem is to find y in each case, we can list the value of y for each row.

Perhaps the expected answer is to fill in the missing y values.

Given that, here are the y values for each row:

Row 1: y = 48
Row 2: y = 3
Row 3: y = -2
Row 4: y = 26
Row 5: y = 10
Row 6: y = 10
Row 7: y = 12
Row 8: y = 10
Row 9: y = 12
Row 10: y = 10

I think that's it.

Final Answer:
48, 3, -2, 26, 10, 10, 12, 10, 12, 10