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Learning Resources

Linear Equations

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 07 Pathway: N/A

First Principles

Objective: Master solving linear equations in one variable using the balance model, maintaining equivalence through inverse operations.

The Golden Rule of Equations: Think of an algebraic equation as a traditional Pan-African twin-pan balance scale (like those used in open-air markets across Nairobi, Lagos, and Accra). The equals sign \(=\) means both pans are in perfect level balance. Whatever you add, subtract, multiply, or divide on the left pan, you must do exactly the same to the right pan to keep the beam level.

Step 1: Concrete Scenario — The Market Balance

Suppose a trader has \(2\) identical sealed sacks of beans (each weighing unknown \(x\) kg) plus a \(5\text{ kg}\) metal counterweight on the left pan. On the right pan sits a \(13\text{ kg}\) weight. The scale is perfectly balanced:

\[ 2x + 5 = 13 \]

To find the weight of one sack \(x\):

  • Step A (Isolate the variable term): Remove the \(5\text{ kg}\) metal weight from the left pan. To keep the beam level, you must also remove \(5\text{ kg}\) from the right pan: \(13 - 5 = 8\text{ kg}\). Now, \(2x = 8\).
  • Step B (Isolate the single unit): The two sacks weigh \(8\text{ kg}\) together. Divide the contents of both pans by \(2\). One sack weighs \(\frac{8}{2} = 4\text{ kg}\).

Step 2: Interactive Balance Simulator

Interactive Beam Balance: \(2x + 5 = 13\)
Left Pan
2x + 5
Right Pan
13

Click the buttons to see the balance stay in equilibrium at each step.

Step 3: The Principle of Inverse Operations

To "undo" an operation and isolate \(x\), apply the inverse (opposite) operation to both sides of the equation:

  • Addition \((+)\) is undone by Subtraction \((-)\).
  • Subtraction \((-)\) is undone by Addition \((+)\).
  • Multiplication \((\times)\) is undone by Division \((\div)\).
  • Division \((\div)\) is undone by Multiplication \((\times)\).

Key Formulas

Standard Two-Step Form: \[ ax + b = c \quad (a \neq 0) \] Procedure:
  1. Subtract the constant: \[ ax = c - b \]
  2. Divide by the coefficient: \[ x = \frac{c - b}{a} \]
Variables on Both Sides: \[ ax + b = cx + d \] Procedure:
  1. Collect variable terms to one side: \[ (a - c)x + b = d \]
  2. Subtract constant \(b\): \[ (a - c)x = d - b \]
  3. Isolate \(x\): \[ x = \frac{d - b}{a - c} \quad (a \neq c) \]
Equations with Brackets & Fractions: \[ \frac{x + p}{q} = r \implies x + p = qr \implies x = qr - p \] \[ k(ax + b) = c \implies kax + kb = c \quad \text{(Distributive Property)} \]
Verification Step (Always Check!):

Substitute the calculated value of \(x\) back into the left-hand side (LHS) and right-hand side (RHS). The solution is correct if and only if:

\[ \text{LHS} = \text{RHS} \]

Worked Examples

Example 1 (Easy — 2-Step Equation):

Solve for \(x\): \[ 3x + 7 = 22 \]

  1. Step 1 (Undo Addition): Subtract \(7\) from both sides to isolate the \(3x\) term. \[ 3x + 7 - 7 = 22 - 7 \] \[ 3x = 15 \]
  2. Step 2 (Undo Multiplication): Divide both sides by \(3\). \[ \frac{3x}{3} = \frac{15}{3} \] \[ x = 5 \]
  3. Check: \(3(5) + 7 = 15 + 7 = 22\) (Correct!).
Example 2 (Medium — Variables on Both Sides):

Solve for \(m\): \[ 5m - 8 = 2m + 13 \]

  1. Step 1 (Group variable terms): Subtract \(2m\) from both sides. \[ 5m - 2m - 8 = 2m - 2m + 13 \] \[ 3m - 8 = 13 \]
  2. Step 2 (Group constant terms): Add \(8\) to both sides. \[ 3m - 8 + 8 = 13 + 8 \] \[ 3m = 21 \]
  3. Step 3 (Isolate variable): Divide by \(3\). \[ m = \frac{21}{3} = 7 \]
  4. Check: LHS \(= 5(7) - 8 = 35 - 8 = 27\). RHS \(= 2(7) + 13 = 14 + 13 = 27\). Both sides match!
Example 3 (Hard — Distributive Property & Fractions):

Solve for \(y\): \[ \frac{2(y - 3)}{5} + 4 = 8 \]

  1. Step 1 (Isolate the fraction): Subtract \(4\) from both sides. \[ \frac{2(y - 3)}{5} = 8 - 4 = 4 \]
  2. Step 2 (Eliminate denominator): Multiply both sides by \(5\). \[ 2(y - 3) = 4 \times 5 = 20 \]
  3. Step 3 (Clear brackets or divide): Divide both sides by \(2\). \[ y - 3 = \frac{20}{2} = 10 \]
  4. Step 4 (Final isolation): Add \(3\) to both sides. \[ y = 10 + 3 = 13 \]
  5. Check: \(\frac{2(13 - 3)}{5} + 4 = \frac{2(10)}{5} + 4 = \frac{20}{5} + 4 = 4 + 4 = 8\). Verified!

Common Mistakes

Misconception 1: Performing an operation on one side only

The Mistake: From \(2x + 5 = 13\), writing \(2x = 13\) and getting \(x = 6.5\).

Why it happens: Learners think of "moving" \(5\) across the equal sign as just erasing it from the left.

The Fix: An equation is a balance. If you remove \(5\) from the left pan, you MUST subtract \(5\) from the right pan: \(2x = 13 - 5 = 8\implies x = 4\).

Misconception 2: Confusing signs when transposing terms

The Mistake: In \(4x - 6 = 14\), subtracting \(6\) to get \(4x = 8\).

Why it happens: Forgetting that the inverse of subtraction \((-6)\) is addition \((+6)\).

The Fix: Always use inverse operations: to undo \(-6\), add \(6\) to both sides: \(4x = 14 + 6 = 20 \implies x = 5\).

Misconception 3: Dividing only part of a side

The Mistake: In \(3x + 9 = 15\), dividing only \(3x\) by \(3\) to get \(x + 9 = 5\).

Why it happens: Attempting division before clearing addition/subtraction, and forgetting that division applies to every term.

The Fix: Follow standard order of operations in reverse: isolate variable terms first by subtracting \(9\) (giving \(3x = 6\)), then divide by \(3\) (giving \(x = 2\)).

Real World

Subtract the booking fee from total: \(3x = 3600 - 600 = 3000\)
Divide among the \(3\) students: \(x = \frac{3000}{3} = 1000\)
Each student pays KES 1,000.

Practice

Baraka the baker harvested bags of wheat. After donating 8 bags to a local primary school feeding program, exactly 42 bags remain in the store. How many bags \(x\) did Baraka originally harvest? (Type only the number, e.g., 42)
Review the concepts above.
A community library has 5 shelves, each holding the same number of books \(x\). After 12 books are borrowed by students, 38 books remain on the shelves. Find the number of books \(x\) originally on each shelf by solving \(5x - 12 = 38\). (Type only the number, e.g., 42)
Review the concepts above.
Halima runs a fruit stall at the market. Mangoes cost KES 6 each. The wholesaler gives her a loyalty discount of KES 12 off her total bill. If Halima paid KES 48 in total, how many mangoes did she buy? Solve \(6x - 12 = 48\). (Type only the number, e.g., 42)
Review the concepts above.
Farmer Baraka harvested 9 equal sacks of wheat, each containing \(x\) kilograms. After selling 36 kg to neighbours, he has 63 kg left. Find the number of kilograms \(x\) in each sack by solving \(9x - 36 = 63\). (Type only the number, e.g., 42)
Review the concepts above.
A hardware supplier in Kisumu sells cement bags at KES 45 each. For a school construction project, they offer a flat coupon discount of KES 120 on the bulk order. If the contractor pays KES 1,230, how many bags of cement were purchased? Solve \(45x - 120 = 1230\). (Type only the number, e.g., 42)
Review the concepts above.
A fruit cooperative packs mangoes into 5 identical crates. Each crate initially contains \(x\) mangoes. Before sealing the crates, quality control removes 3 damaged mangoes from each crate. In total, 35 good mangoes are packed across all 5 crates: \(5(x - 3) = 35\). How many mangoes were originally in each crate? (Type only the number, e.g., 42)
Review the concepts above.