Practice Maths

Solving Linear Equations II

Backtracking to Solve Equations

Key Terms

function machine
a model where an input value is transformed by a rule to give an output
forward path
the sequence of operations applied to x to build the equation
backward path
the reverse sequence of operations used to find x
backtracking
solving an equation by reversing each operation in reverse order

The Function Machine (Forward)

Think of an equation as a machine. You put a number in (Input x), the machine applies rules, and spits a number out (Output y).

Example: The machine rule is y = 3x.

  • Input 5.
  • Machine calculates 3 × 5.
  • Output is 15.

Backtracking (Reverse)

Sometimes we know the Output, but we want to find the Input. We have to work backwards using Inverse Operations.

Example: Solve 2x − 5 = 11.

Forward Path (Building the equation) Backward Path (Solving the equation)
Start with x Start with 11
Multiply by 2 (× 2) Add 5 (+ 5)
Subtract 5 (− 5) Divide by 2 (÷ 2)
End with 11 End with x

Algebraic Solution:

  • 11 + 5 = 16
  • 16 ÷ 2 = 8
  • So, x = 8.
Hot Tip Order of operations (BODMAS) matters! When backtracking, you must peel away the operations in the reverse order they were applied.

Worked Example

Problem: Solve for x: 2x + 5 = 13

Step 1 — Identify the operations and work backwards (backtrack).
Forward path: Start with x, multiply by 2, then add 5, to get 13.
Backward path: Start with 13, subtract 5, then divide by 2.

Step 2 — Apply the inverse operations in reverse order.
13 − 5 = 8
8 ÷ 2 = 4
So x = 4.

Step 3 — Check by substituting back into the original equation.
2(4) + 5 = 8 + 5 = 13. ✓

The Function Machine Idea

Imagine a machine at a factory. A number goes in one end (the input), the machine does something to it, and a result comes out the other end (the output). The "something" the machine does is your equation.

For y = 3x: put in 5, the machine multiplies by 3, and out comes 15. Simple! But what if the machine spat out 15, and you need to know what went IN? You have to reverse the machine — that's backtracking!

Building the Forward and Backward Path

Let's work through 2x − 5 = 11 using the Key Ideas table method:

Forward path (how the equation was built):

  • Start with x
  • Multiply by 2
  • Subtract 5
  • Arrive at 11

Backward path (how we solve it — reverse every step):

  • Start with 11
  • Add 5 (undoes "subtract 5"): 11 + 5 = 16
  • Divide by 2 (undoes "multiply by 2"): 16 ÷ 2 = 8
  • Arrive at x = 8

Verify: 2(8) − 5 = 16 − 5 = 11. Correct!

Why "Reverse BODMAS" Matters

BODMAS tells us to work from inside brackets, then powers, then multiplication/division, then addition/subtraction. When we undo an equation, we go in the opposite order: deal with addition/subtraction first, then multiplication/division last.

Think of getting dressed vs undressed. If you put on socks then shoes, you take off shoes first, then socks. Reverse the order! Equations work the same way.

Remember: When backtracking, undo the LAST thing that happened to x first. In 3x + 2 = 14, the last thing was "add 2" — so undo that first (subtract 2), then undo "multiply by 3" (divide by 3).

Brackets: An Extra Layer

The equation 3(m + 4) = 27 looks scarier but the logic is identical. The forward path here is: "add 4 to m, then multiply by 3." So the backward path is: "divide by 3 first, then subtract 4."

  • 27 ÷ 3 = 9 (undo "multiply by 3")
  • 9 − 4 = 5 (undo "add 4")
  • m = 5

Check: 3(5 + 4) = 3(9) = 27. Correct!

Common Mistake: Students try to expand brackets first (multiply 3 by m and 3 by 4). That's a valid method too, but backtracking is much faster for one-variable equations. Choose whichever approach feels clearest to you.

Think Like a Detective

Solving an equation is really detective work. The answer (output) is the clue. You work backwards through the crime scene (the operations) to find the culprit (the value of x). Every step you take is justified by a mathematical rule — just like a detective needs evidence for each conclusion.

Practice Questions

  1. Forward Substitution Fluency

    Calculate the output (y) for the following inputs:

    1. Function: y = 4x + 2. Input x = 5.
    2. Function: y = 10 − 2x. Input x = 3.
    3. Function: y = 3(x + 2). Input x = 4.

  2. Simple Backtracking Fluency

    A function machine is set to y = x − 5.

    1. If the output is 20, what was the input?
    2. If the output is 8, what was the input?

  3. Identifying Operations in Equations Fluency

    For each equation, state the operation applied to x and the inverse operation needed to isolate x. Then solve.

    1. x + 8 = 15
    2. 3x = 21
    3. x − 4 = 9
    4. x ÷ 5 = 6

  4. Mapping the Path Understanding

    Consider the equation 3x + 2 = 14.

    1. Forward: Start with x. What is the first operation applied to x?
    2. Forward: What is the second operation?
    3. Backward: To solve it, what is the first operation you must do to 14?
    4. Backward: What is the second operation?

  5. Solving One-Step Equations (Multiplication) Fluency

    Use inverse operations to solve for the variable:

    1. 4m = 36
    2. 6y = 48

  6. Solving One-Step Equations (Addition and Subtraction) Fluency

    Use inverse operations to solve:

    1. k + 9 = 21
    2. p − 7 = 15

  7. Solving Two-Step Equations Understanding

    Use backtracking to solve:

    1. 3p − 4 = 17
    2. 2x + 9 = 21

  8. More Two-Step Equations Understanding

    Solve the following:

    1. 5y + 10 = 45
    2. 4m − 1 = 23

  9. Solving with Brackets Understanding

    Consider the equation: 3(m + 4) = 27.

    Note: The forward path is "Add 4, then Multiply by 3".

    1. What is the first step in the reverse (backtracking) path?
    2. Solve for m.

  10. Word Problem Problem Solving

    I think of a number. I multiply it by 2, and then add 3. The result is 15.

    1. Write the equation (use n for the number).
    2. Use backtracking to find the number.
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