Formula Rearrangement

Grade 9 · algebra · 100 practice problems · read aloud

🔊 Listen to this explanation

Formula Rearrangement: The Art of Solving for Any Variable

🔍 What is Formula Rearrangement?

Formula rearrangement is the process of manipulating an equation to isolate a specific variable. Instead of just solving for x, you're reorganizing the entire formula so that any variable can be the subject. This is super useful in science (like physics and chemistry) and real-life problems where you need to find different quantities.

📝 Step-by-Step Guide

  1. Identify the target variable you want to solve for.
  2. Use inverse operations to "undo" what's being done to the target variable. Perform the same operation to both sides of the equation to keep it balanced.
  3. Simplify both sides of the equation after each step.
  4. Check your work by plugging your final formula back into the original equation with sample numbers.

✨ Visual Examples

Example 1: Rearrange the area of a rectangle formula for width (w).

Original: A = l × w

Step 1: We want w alone. Since w is multiplied by l, we do the inverse: divide both sides by l.

A / l = (l × w) / l

Step 2: Simplify.

w = A / l

Example 2: Rearrange the linear equation for x.

Original: y = 3x - 5

Step 1: Undo subtraction. Add 5 to both sides: y + 5 = 3x

Step 2: Undo multiplication. Divide both sides by 3: (y + 5) / 3 = x

x = (y + 5) / 3

⚠️ Common Mistakes

  • Forgetting to perform the operation on both sides: Whatever you do to one side, you MUST do to the other.
  • Incorrect order of operations: Work in reverse BEDMAS order. Undo addition/subtraction before multiplication/division.
  • Misplacing negative signs: Be extra careful when moving terms that are negative.

💡 Tips & Tricks

  • Think of the equation as a balanced scale. To keep it balanced, you must add/remove the same weight from both sides.
  • Use the "Golden Rule of Algebra": Do unto one side as you do unto the other.
  • If the variable is in the denominator (e.g., 5/t), multiply both sides by that variable first to "bring it up."

🎯 Practice Suggestions

To master this skill, try these activities:

  • Start with simple formulas like A = lw or C = πd and rearrange for each variable.
  • Practice with common science formulas like F = ma (solve for m, then for a) or v = d/t.
  • Create flashcards with a formula on the front and the rearranged versions on the back.
  • Check your answers by substituting numbers into both the original and your new formula to see if they give the same result.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 Solve for h: V = πr²h

Hint: Think about inverse operations. The formula multiplies π, r², and h together. To isolate h, perform the opposite operation on both sides of the equation.

Show the answer

Answer: h = V/(πr²)

  1. Start with the formula V = πr²h.
  2. To isolate h, divide both sides by πr² (since πr² is multiplied by h).
  3. V / (πr²) = (πr²h) / (πr²)
  4. Simplify the right side: πr² cancels, leaving h.
  5. Therefore, h = V / (πr²).

The answer is h = V/(πr²).

2 Solve for r: S = 4πr²

Hint: Isolate the squared term by dividing both sides by the coefficient, then take the square root to solve for the variable.

Show the answer

Answer: r = sqrt(S / (4π))

  1. Start with S = 4πr²
  2. Divide both sides by 4π to isolate r²: S / (4π) = r²
  3. Take the square root of both sides: r = sqrt(S / (4π))

The answer is r = sqrt(S / (4π)).

3 Solve for h: V = πr²h/3

Hint: Think about which operations are being applied to h and reverse them in the opposite order.

Show the answer

Answer: h = 3V/(πr²)

  1. Start with the formula V = (πr²h)/3.
  2. Multiply both sides by 3 to eliminate the denominator: 3V = πr²h.
  3. Divide both sides by πr² to isolate h: h = 3V/(πr²).

The answer is h = 3V/(πr²).

4 Solve for r: V = (4/3)πr³

Hint: Think about inverse operations. To undo a cube, you take the cube root. Work backwards from the formula, isolating the variable step by step.

Show the answer

Answer: r = ∛(3V/(4π))

  1. Start with V = (4/3)πr³
  2. Multiply both sides by 3 to eliminate the fraction: 3V = 4πr³
  3. Divide both sides by 4π to isolate r³: 3V/(4π) = r³
  4. Take the cube root of both sides to solve for r: r = ∛(3V/(4π))

The answer is r = ∛(3V/(4π)).

5 Solve for t: d = 4t² + 8t

Hint: This is a quadratic equation in t. Rearrange it into standard form and use the quadratic formula to solve for t.

Show the answer

Answer: t = (-2 + sqrt(4 + d)) / 2 or t = (-2 - sqrt(4 + d)) / 2

  1. Start with the equation d = 4t² + 8t.
  2. Rearrange into standard quadratic form: 4t² + 8t - d = 0.
  3. Use the quadratic formula t = [-b ± sqrt(b² - 4ac)] / (2a), where a = 4, b = 8, c = -d.
  4. Substitute: t = [-8 ± sqrt(8² - 4*4*(-d))] / (2*4) = [-8 ± sqrt(64 + 16d)] / 8.
  5. Simplify the square root: sqrt(64 + 16d) = sqrt(16(4 + d)) = 4 sqrt(4 + d).
  6. So t = [-8 ± 4 sqrt(4 + d)] / 8.
  7. Divide numerator and denominator by 4: t = [-2 ± sqrt(4 + d)] / 2. The solutions are t = (-2 + sqrt(4 + d)) / 2 and t = (-2 - sqrt(4 + d)) / 2.

6 Solve for h: V = (1/3)πr²h

Hint: Think about inverse operations. The variable h is multiplied by other terms, so undo those multiplications one at a time, starting with the fraction 1/3.

Show the answer

Answer: h = 3V/(πr²)

  1. Start with V = (1/3)πr²h
  2. Multiply both sides by 3 to eliminate the fraction: 3V = πr²h
  3. Divide both sides by πr² to isolate h: h = 3V/(πr²)

The answer is h = 3V/(πr²).

Practise this topic — 10 free problems, no signup →