Algebra 1 Worksheets | Equations Worksheets - Free Printable

Let’s solve each equation step by step. We’ll simplify both sides, move variables to one side and constants to the other, then solve for the variable. For equations 7–12, we’ll also check if there’s one solution, no solution, or infinite solutions.

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1) -68 + 30k = 10(3k - 7) + 2

First, expand the right side:
10 × 3k = 30k
10 × (-7) = -70
So right side becomes: 30k - 70 + 2 = 30k - 68

Now equation is:
-68 + 30k = 30k - 68

Subtract 30k from both sides:
-68 = -68 → This is always true!

✔ Infinite solutions (any value of k works)

But wait — the instruction says “Solve each equation” for 1–6. Since it’s an identity, we can say “all real numbers” or just note it’s always true. But let’s keep going as instructed.

Actually, since it simplifies to a true statement with no variable left, it has infinite solutions.

But maybe they want us to write that? Let’s hold on — perhaps for 1–6, even if infinite, we still report it.

Wait — looking back at instructions: “Solve each equation.” for 1–6, and for 7–12, specifically state type of solution.

So for 1–6, we should solve normally. If it reduces to identity, we say infinite solutions. If contradiction, no solution.

But let’s do them all carefully.

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1) -68 + 30k = 10(3k - 7) + 2

Right side: 10*(3k - 7) + 2 = 30k - 70 + 2 = 30k - 68

Left side: -68 + 30k

So: -68 + 30k = 30k - 68 → same on both sides → subtract 30k: -68 = -68 → TRUE

→ Infinite solutions

But maybe the problem expects us to write “all real numbers” or something. Let’s proceed.

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2) 48(m + 3) - 102 = 6(8m + 7)

Expand both sides:

Left: 48*m + 48*3 - 102 = 48m + 144 - 102 = 48m + 42

Right: 6*8m + 6*7 = 48m + 42

So: 48m + 42 = 48m + 42 → again, identical → subtract 48m: 42 = 42 → TRUE

→ Infinite solutions

Hmm, interesting. Let’s continue.

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3) 21 + 3h = 3(h + 7)

Right side: 3*h + 3*7 = 3h + 21

Left: 21 + 3h

So: 21 + 3h = 3h + 21 → same → subtract 3h: 21 = 21 → TRUE

→ Infinite solutions

This pattern continues? Let’s check #4.

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4) 4 = -8z - 7 + 8z

Combine like terms on right: -8z + 8z = 0, so right side = -7

Equation: 4 = -7 → FALSE!

→ No solution

Ah, here’s a contradiction.

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5) -8n + 7 + 8n = 7

Combine like terms: -8n + 8n = 0, so left side = 7

Equation: 7 = 7 → TRUE

→ Infinite solutions

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6) 3 + 4v = 6(5 - 7v)

Expand right side: 6*5 = 30, 6*(-7v) = -42v → 30 - 42v

Equation: 3 + 4v = 30 - 42v

Add 42v to both sides: 3 + 46v = 30

Subtract 3: 46v = 27

Divide by 46: v = 27/46

Check if reducible: 27 and 46 share no common factors → leave as fraction.

→ One solution: v = 27/46

Okay, now problems 7–12 ask to state whether one, infinite, or no solutions, and if one, solve.

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7) 2(-5f - 9) = -22 - 10f

Left side: 2*(-5f) + 2*(-9) = -10f - 18

Equation: -10f - 18 = -22 - 10f

Add 10f to both sides: -18 = -22 → FALSE

→ No solution

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8) -66 = 9(-7 - 8r) + 72r

Expand right: 9*(-7) = -63, 9*(-8r) = -72r → so -63 -72r + 72r = -63 + 0r = -63

Equation: -66 = -63 → FALSE

→ No solution

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9) 6(f - 10) = -60 + 6f

Left: 6f - 60

Right: -60 + 6f → same as 6f - 60

So: 6f - 60 = 6f - 60 → subtract 6f: -60 = -60 → TRUE

→ Infinite solutions

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10) 12 = -6(9t - 2) + 54t

Expand right: -6*9t = -54t, -6*(-2) = +12 → so -54t + 12 + 54t = 12 + 0t = 12

Equation: 12 = 12 → TRUE

→ Infinite solutions

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11) 8(5 - 4c) + 25c + 7c = 10

First, expand: 8*5 = 40, 8*(-4c) = -32c

So: 40 - 32c + 25c + 7c = 10

Combine c terms: -32c + 25c = -7c; -7c + 7c = 0c

So: 40 + 0c = 10 → 40 = 10 → FALSE

→ No solution

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12) 8 + u = 5u - 4(u + 9)

First, expand right: -4*u = -4u, -4*9 = -36 → so 5u -4u -36 = u - 36

Equation: 8 + u = u - 36

Subtract u from both sides: 8 = -36 → FALSE

→ No solution

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Now, let’s summarize answers clearly.

For problems 1–6, even though some have infinite or no solutions, we solved them as per instruction.

But note: In problem 1,2,3,5 → infinite solutions
Problem 4 → no solution
Problem 6 → one solution: v=27/46

For 7–12, we stated type and solved if one solution.

Only problem 6 had one solution among 1–6.

Let me double-check problem 6:

3 + 4v = 6(5 - 7v)
3 + 4v = 30 - 42v
4v + 42v = 30 - 3
46v = 27
v = 27/46 → correct.

And others are identities or contradictions.

Now, final answers:

──────────────────────────────────────

Final Answer:
1) Infinite solutions
2) Infinite solutions
3) Infinite solutions
4) No solution
5) Infinite solutions
6) v = 27/46
7) No solution
8) No solution
9) Infinite solutions
10) Infinite solutions
11) No solution
12) No solution