Add equations: 8x = 16, x = 2, y = 3.

Solve Using Elimination Method — 3x + 2y = 12, 5x - 2y = 4

Question

Solve the following pair of linear equations using the elimination method:

3x + 2y = 12 \quad \text{...(1)}

5x - 2y = 4 \quad \text{...(2)}

Solution — Step by Step

Look at the $y$ terms: equation (1) has $+2y$ and equation (2) has $-2y$ . The coefficients are equal in magnitude and opposite in sign. This means if we simply add the two equations, $y$ vanishes completely. No multiplication needed — the question is already set up beautifully.

3x + 2y = 12

5x - 2y = 4

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8x + 0y = 16

The $y$ terms cancel. We now have a single-variable equation: $8x = 16$ .

8x = 16 \implies x = 2

Use equation (1) — it’s slightly simpler with smaller numbers:

3(2) + 2y = 12

6 + 2y = 12

2y = 6 \implies y = 3

You can substitute into either equation. Always prefer the one that looks less messy.

Always verify — CBSE awards 1 mark specifically for this step in many cases.

Check in (1): $3(2) + 2(3) = 6 + 6 = 12$ ✓
Check in (2): $5(2) - 2(3) = 10 - 6 = 4$ ✓

Answer: $x = 2,\ y = 3$

Why This Works

Elimination works on a simple idea: if two equal quantities are added to two sides of an equation, the equality holds. When we add equation (1) and equation (2), we’re adding the same “amount” ( $12 + 4 = 16$ ) to both sides, just written differently.

The trick is making one variable’s coefficient become zero after addition or subtraction. Here, $+2y$ and $-2y$ are already opposites, so adding is all it takes. In harder problems, we multiply one or both equations by a constant first to create this situation.

Think of it as engineering a cancellation — we’re choosing our operation to knock out one variable so the other becomes easy to isolate.

Alternative Method — Substitution

From equation (1), express $x$ in terms of $y$ :

3x = 12 - 2y \implies x = \frac{12 - 2y}{3}

Substitute into equation (2):

5 \cdot \frac{12 - 2y}{3} - 2y = 4

\frac{60 - 10y}{3} - 2y = 4

Multiply through by 3:

60 - 10y - 6y = 12

60 - 16y = 12

16y = 48 \implies y = 3

Then $x = \frac{12 - 2(3)}{3} = \frac{6}{3} = 2$ .

Same answer. Substitution works, but elimination was clearly faster here — whenever coefficients are already matched, always prefer elimination.

In CBSE boards, if the question says “use elimination method”, you must show the elimination steps explicitly. Switching to substitution midway — even if correct — can cost you method marks.

Common Mistake

Many students add the equations correctly to get $8x = 16$ , find $x = 2$ , and then substitute back into the wrong equation carelessly — or worse, make a sign error during substitution and get $y = -3$ . Always write out the substitution step fully. Don’t do it in your head under exam pressure. One line of working costs nothing; a wrong answer costs marks.

A second trap: when the question gives $5x - 2y = 4$ , some students rewrite $-2y$ as $+2y$ while copying — a copying error that changes the entire problem. Box the signs when you write down the equations at the start.

Question

Solution — Step by Step

Why This Works

Alternative Method — Substitution

Common Mistake

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