Multi-Step Equations Practice Worksheet featuring angle problems with diagrams and equation solving.

Let’s solve each problem one by one. We’ll look at the angles, write an equation based on what we know about angle relationships, and then solve for x.

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Problem 1:

We see two angles that form a straight line — they are supplementary (add up to 180°).

One angle is labeled “6x - 37” and the other is 147°.

So:

> (6x - 37) + 147 = 180

Simplify:

> 6x + 110 = 180
> 6x = 70
> x = 70 ÷ 6
> x = 35/3 ≈ 11.67 → But let’s keep it as a fraction unless told otherwise.

Wait — let me double-check:

(6x - 37) + 147 = 180
→ 6x + (147 - 37) = 180
→ 6x + 110 = 180
→ 6x = 70
→ x = 70/6 = 35/3

But maybe I made a mistake? Let’s plug back in:

If x = 35/3, then 6x = 70, so 70 - 37 = 33. Then 33 + 147 = 180 ✔

Okay, correct.

But wait — looking again at the diagram: the angle labeled “6x - 37” and the 147° angle are adjacent and form a straight line — yes, supplementary.

✔ So x = 35/3 or approximately 11.67. But since this is likely expecting exact value, we leave as fraction.

Actually — hold on! Maybe I misread the diagram. Is the 147° the whole straight angle? Or is it part of it?

Looking again: The arrow shows the full straight line is split into two parts: one is “6x - 37”, the other is 147°? That can’t be right because 147° alone is already more than half of 180°, and if you add another positive angle, it would exceed 180°.

Wait — perhaps the 147° is the measure of the *other* angle? No — actually, re-examining: the diagram probably means that the two angles together make 180°, and one is labeled “6x - 37”, and the other is NOT labeled with a number — but wait, no — in the image description, it says “147°” is written next to one ray.

Actually, I think there's confusion here. Let me reinterpret.

In many such diagrams, when two rays form a straight line, and one angle between them is labeled with an expression and the other with a number, they are supplementary.

But if one angle is 147°, and the other is “6x - 37”, then:

6x - 37 + 147 = 180 → same as before.

But 6x - 37 must be the smaller angle? 147° is large, so 6x - 37 should be 33°, which makes sense.

So x = 35/3 is correct.

But let’s check Problem 2 to see pattern.

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Problem 2:

Angle XYZ is given as 80°. It’s composed of two smaller angles: one is 5x + 9, the other is 2x + 1.

Since they are adjacent and together make angle XYZ, we add them:

> (5x + 9) + (2x + 1) = 80
> 7x + 10 = 80
> 7x = 70
> x = 10

Check: 5(10)+9=59, 2(10)+1=21, 59+21=80 ✔

Good.

Back to Problem 1 — maybe I misread. Perhaps the 147° is the total? No, the label “147°” is placed on one side.

Alternative interpretation: Maybe the angle labeled “6x - 37” and the 147° are vertical angles? But they don’t look like it — they’re on a straight line.

Another possibility: In some diagrams, the number outside might indicate the reflex angle? Unlikely.

Perhaps the 147° is the measure of the angle opposite? But no indication.

Wait — let’s assume my first calculation is correct: x = 35/3.

But let’s move on and come back.

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Problem 3:

Two lines intersect. One angle is labeled “x”, the adjacent angle is 128°. Since they form a straight line, they are supplementary.

So:

> x + 128 = 180
> x = 52

Also, vertically opposite angles are equal, but here we only need the linear pair.

✔ x = 52

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Problem 4:

Right angle symbol — so the two angles add to 90°.

Angles: 5x + 17 and 38°

So:

> (5x + 17) + 38 = 90
> 5x + 55 = 90
> 5x = 35
> x = 7

Check: 5(7)+17=35+17=52; 52+38=90 ✔

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Problem 5:

Vertical angles? Or linear pair?

The diagram shows two lines crossing. One angle is labeled “2x + 71”, and the opposite angle is not labeled, but there’s a 47° angle adjacent? Wait — actually, looking at typical setup: often, when two lines cross, vertical angles are equal.

Here, it seems that “2x + 71” and the 47° angle are vertical angles? Because they are opposite each other.

Yes — in intersecting lines, vertical angles are equal.

So:

> 2x + 71 = 47
> 2x = 47 - 71 = -24
> x = -12

Negative? Possible mathematically, but let’s verify.

If x = -12, then 2(-12) + 71 = -24 + 71 = 47 ✔

So even though negative, it’s correct.

Alternatively, could they be supplementary? If they were adjacent, but the diagram suggests they are vertical.

Given the layout, vertical angles make sense.

✔ x = -12

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Problem 6:

Intersecting lines. Angles: one is 47°, another is 3x - 8, and they appear to be vertical angles? Or adjacent?

Looking: typically, in such diagrams, if two angles are opposite, they are vertical.

Here, 47° and 3x - 8 seem to be vertical angles.

So:

> 3x - 8 = 47
> 3x = 55
> x = 55/3 ≈ 18.333...

Check: 3*(55/3) - 8 = 55 - 8 = 47 ✔

So x = 55/3

But let’s confirm the diagram: if they are vertical, yes.

If they were adjacent, they’d sum to 180, but 47 + (3x-8) = 180 → 3x = 141 → x=47, then 3*47-8=133, 47+133=180 — also possible.

Which is it?

In standard notation, when two lines cross, and angles are labeled on opposite sides, they are vertical.

In Problem 5, we assumed vertical for 2x+71 and 47, got x=-12, which worked.

Similarly here, if 3x-8 and 47 are vertical, then equal.

But let’s see the position: in Problem 6, the 47° is on one side, 3x-8 on the opposite — likely vertical.

To decide: in Problem 5, if we had assumed supplementary, 2x+71 + 47 = 180 → 2x=62 → x=31, then 2*31+71=62+71=133, and 133+47=180 — also works.

But which relationship is intended?

This is ambiguous without seeing the exact diagram.

However, in most textbooks, when an angle is labeled directly opposite another, it’s vertical.

Moreover, in Problem 5, if x=31, then 2x+71=133, and if 47 is adjacent, then yes, but the diagram might show them as vertical.

Given that in Problem 2 and 4, we used addition for adjacent angles, and in Problem 3, supplementary, here for intersecting lines, vertical angles are equal.

I think for consistency, in Problems 5 and 6, the labeled angles are vertical.

So for Problem 6: 3x - 8 = 47 → x = 55/3

But let’s note that 55/3 is approximately 18.333, which is fine.

Now back to Problem 1.

In Problem 1, if the two angles are on a straight line, and one is 147°, the other must be 33°, so 6x - 37 = 33 → 6x = 70 → x = 35/3

Same as before.

Perhaps the answer is expected as fraction.

Maybe I misread the diagram: is 147° the measure of the angle that includes both? Unlikely.

Another thought: sometimes the number is written outside to indicate the larger angle, but here it's placed on one segment.

I think we have to go with x = 35/3 for Problem 1.

But let’s list all answers:

Problem 1: x = 35/3
Problem 2: x = 10
Problem 3: x = 52
Problem 4: x = 7
Problem 5: x = -12
Problem 6: x = 55/3

For Problem 5, x=-12 might be acceptable, but let’s double-check the diagram assumption.

In Problem 5, if the angles are vertical, then equal. If they are adjacent, supplementary.

In the description, it says "two lines crossing", and "one angle is 2x+71", and "another is 47°". Typically, if they are opposite, vertical; if next to each other, supplementary.

Without the image, it's hard, but in many worksheets, when two angles are labeled on opposite sides of intersection, they are vertical.

Similarly for Problem 6.

To resolve, let's consider that in Problem 3, we have a clear linear pair: x and 128° add to 180.

In Problem 1, similarly, linear pair.

In Problems 5 and 6, intersecting lines, so vertical angles are equal.

I think that's standard.

So final answers:

1. x = 35/3
2. x = 10
3. x = 52
4. x = 7
5. x = -12
6. x = 55/3

But let's write them as fractions where needed.

For the worksheet, they might want simplified fractions or decimals, but since not specified, fractions are exact.

Now, for the response, since the user didn't ask for explanation per problem, but to solve, and the options are false, I just need to provide the final answer.

But the instruction says: "Complete ALL reasoning and explanations BEFORE writing the Final Answer."

And since ExplainAnswer is false, I shouldn't explain, but I did reason above.

In the response format, if ExplainAnswer is false, omit that section.

Similarly for others.

So I will provide only the Final Answer section with the solutions.

But the Final Answer should be the answers to the problems.

How to present? Probably list them.

Since it's a worksheet with 6 problems, I'll list the solution for each.

Final Answer:

1. \( x = \frac{35}{3} \)
2. \( x = 10 \)
3. \( x = 52 \)
4. \( x = 7 \)
5. \( x = -12 \)
6. \( x = \frac{55}{3} \)

To box them or something, but since it's multiple, list clearly.

Perhaps write as:

For problem 1: x = 35/3
For problem 2: x = 10
etc.

But in the context, since it's a single response, I'll write:

Final Answer:
Problem 1: \( \frac{35}{3} \)
Problem 2: \( 10 \)
Problem 3: \( 52 \)
Problem 4: \( 7 \)
Problem 5: \( -12 \)
Problem 6: \( \frac{55}{3} \)

Yes.