Why bother with substitution and elimination?
Imagine you have two riddles that hide the same two secret numbers. Solving them together is like unlocking a treasure chest – you get both answers in one go.
💡 In Simple Words: When two straight‑line equations share the same x and y, you can find the exact values of x and y by either plugging one equation into the other (substitution) or by adding/subtracting them to cancel a variable (elimination).
What is a pair of linear equations?
A linear equation is an algebraic sentence where the highest power of the variable is one – think of a straight line on a graph. A pair simply means two such equations written together, usually like:
ax + by = c
dx + ey = f
Here, a, b, c, d, e, f are numbers, and x and y are the unknowns we need to discover.
Substitution method – step by step
Substitution is like solving a puzzle by first isolating one piece, then fitting it into the other picture.
Steps
- Pick the simpler equation (the one with smaller coefficients or a single variable).
- Rewrite it so that one variable is alone on one side – this is called isolating the variable.
- Plug that expression into the other equation wherever the same variable appears.
- Solve the resulting single‑variable equation.
- Back‑substitute the found value into the expression from step 2 to get the second variable.
Worked example
Solve the pair:
2x + 3y = 12
x - y = 1
Step 1: The second equation looks easier. Isolate x:
x = y + 1
Step 2: Substitute x in the first equation:
2(y + 1) + 3y = 12
Expand: 2y + 2 + 3y = 12 → 5y + 2 = 12
Step 3: Solve for y:
5y = 10 → y = 2
Step 4: Back‑substitute y = 2 into x = y + 1:
x = 2 + 1 = 3
Solution: (x, y) = (3, 2). Quick check: 2·3 + 3·2 = 6 + 6 = 12 ✓
Elimination method – step by step
Elimination works like a tug‑of‑war: you add or subtract the equations so that one variable “drops out” (cancels) and you’re left with a single‑variable equation.
Steps
- Look at the coefficients (the numbers in front of x or y). Choose a variable whose coefficients can be made equal (or opposite) by multiplying one or both equations.
- Multiply the whole equations as needed.
- Add or subtract the equations so that the chosen variable disappears.
- Solve the remaining single‑variable equation.
- Plug that value back into one of the original equations to find the other variable.
Worked example
Solve:
3x + 4y = 24
5x - 4y = 2
Step 1: Notice the y‑coefficients are +4 and –4 – they already cancel if we add the equations.
Step 2: Add the two equations:
(3x + 5x) + (4y – 4y) = 24 + 2 → 8x = 26
Step 3: Solve for x:
x = 26/8 = 13/4 = 3.25
Step 4: Substitute x back into the first equation:
3(3.25) + 4y = 24 → 9.75 + 4y = 24 → 4y = 14.25 → y = 3.5625
Solution: (x, y) = (13/4, 57/16). Verify with the second equation – it works.
Quick comparison – when to use which?
| Method | Best for | Pros | Cons |
|---|---|---|---|
| Substitution | One equation already has a single variable or easy coefficient | Fewer arithmetic steps if isolation is simple | Can create fractions early, making calculations messy |
| Elimination | Coefficients can be matched easily, especially with opposite signs | Handles whole numbers nicely, avoids early fractions | May require extra multiplication of whole equations |
Exam‑ready tips
- Always write down what you are solving for – x and y – before you start.
- Check if any coefficient is 1; that usually signals a good candidate for substitution.
- When using elimination, aim to cancel the variable with the smallest absolute coefficient to keep numbers small.
- After finding one variable, substitute back into the original (not the modified) equation to avoid propagation of rounding errors.
- Do a quick sanity check by plugging both values into each original equation.
📝 Likely Exam Questions
- Solve by substitution: 4x – 5y = 7 and x + 2y = 5.
Answer: x = 3, y = 1. - Solve by elimination: 6x + 2y = 20 and 3x – 4y = -2.
Answer: x = 2, y = 4. - Explain why elimination is preferable when the coefficients of y are opposites.
Answer: Adding the equations cancels y instantly, leaving a simple equation in x. - Given 2x + 3y = 18 and 4x + 6y = 36, determine the nature of the system.
Answer: The equations are multiples of each other, so infinitely many solutions (dependent system). - Use either method to solve: 5x – y = 9 and 2x + 3y = 4.
Answer: x = 2, y = 1.