Literal Equations Worksheets - Math Monks - Free Printable

Let’s solve each problem step by step.

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Problem 1: 6x = 18, for x

We want to get x alone on one side.
Since x is multiplied by 6, we divide both sides by 6:

6x ÷ 6 = 18 ÷ 6
→ x = 3

✔ Check: 6 × 3 = 18 → Correct!

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Problem 2: A = (1/2)h(a + b), for a

We need to isolate “a”.

Step 1: Multiply both sides by 2 to eliminate the fraction:

2A = h(a + b)

Step 2: Divide both sides by h:

(2A)/h = a + b

Step 3: Subtract b from both sides:

(2A)/h - b = a

So, a = (2A)/h - b

✔ Check: Plug back in — if a = (2A)/h - b, then a + b = (2A)/h → multiply by h → h(a+b) = 2A → half of that is A → matches original equation. Good!

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Problem 3: y = (f + g)/3, for g

Multiply both sides by 3:

3y = f + g

Subtract f from both sides:

3y - f = g

So, g = 3y - f

✔ Check: If g = 3y - f, then f + g = f + 3y - f = 3y → divided by 3 is y → correct.

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Problem 4: 3ax + b = c, for x

Subtract b from both sides:

3ax = c - b

Divide both sides by 3a:

x = (c - b)/(3a)

✔ Check: Plug back in — 3a × [(c-b)/(3a)] + b = (c-b) + b = c → correct.

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Problem 5: v = (1/3)Bh, for h

Multiply both sides by 3:

3v = Bh

Divide both sides by B:

h = (3v)/B

✔ Check: (1/3)B × (3v/B) = (1/3)(3v) = v → correct.

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Problem 6: y - y₁ = m(x - x₁), for x

Add y₁ to both sides:

y = m(x - x₁) + y₁ → but better to keep as:

y - y₁ = m(x - x₁)

Divide both sides by m:

(y - y₁)/m = x - x₁

Add x₁ to both sides:

x = (y - y₁)/m + x₁

✔ Check: Plug into right side: m[ ((y-y₁)/m + x₁) - x₁ ] = m[(y-y₁)/m] = y - y₁ → matches left side. Good.

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Problem 7: s = 2(lw + lh + wh), for w

First, divide both sides by 2:

s/2 = lw + lh + wh

Group terms with w:

s/2 = w(l + h) + lh

Subtract lh from both sides:

s/2 - lh = w(l + h)

Now divide both sides by (l + h):

w = (s/2 - lh) / (l + h)

You can also write it as:

w = [s - 2lh] / [2(l + h)] ← multiplying numerator and denominator by 2 to eliminate fraction inside.

Either form is acceptable, but let’s use the cleaner version:

w = (s - 2lh) / [2(l + h)]

✔ Check: Let’s plug numbers. Suppose l=2, h=3, w=4 → then s = 2(2×4 + 2×3 + 3×4) = 2(8+6+12)=2(26)=52

Now compute w using formula: (52 - 2×2×3) / [2(2+3)] = (52-12)/10 = 40/10 = 4 → correct.

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Problem 8: g = 4ca - 3ba, for a

Factor out ‘a’ on the right:

g = a(4c - 3b)

Now divide both sides by (4c - 3b):

a = g / (4c - 3b)

✔ Check: Plug back — a(4c - 3b) = [g/(4c-3b)] × (4c-3b) = g → correct.

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Final Answers:

1. x = 3
2. a = (2A)/h - b
3. g = 3y - f
4. x = (c - b)/(3a)
5. h = 3v/B
6. x = (y - y₁)/m + x₁
7. w = (s - 2lh) / [2(l + h)]
8. a = g / (4c - 3b)