The elimination method · Code with Martin

Add the equations, an unknown disappears

The second method for simultaneous equations is even faster — when it fits. It is called elimination (adding the equations).

Look at x + y = 8 and x − y = 2. In the first equation there is +y, in the second −y. What happens when you add both equations? Left sides together: x + y + x − y = 2x. The y cancels! Right sides: 8 + 2 = 10. You are left with 2x = 10, so x = 5.

You may do this because an equation is like a scale in balance. When you add the contents of one balanced scale onto another — left pan to left, right pan to right — the balance stays.

Elimination is strongest when opposite numbers stand by one unknown: +y and −y, +3x and −3x. When they are not there, you can make them — you multiply one equation by a suitable number.

In entrance exams it helps to have both methods in your hand and pick according to the question. Today you add the second one.

Adam and Tereza race who solves x + y = 15 and x − y = 3 first. Tereza goes by substitution: make it the subject, substitute, expand… Adam just adds the equations under each other: “2x = 18, x = 9, y is 15 minus 9, so 6. Done.” Tereza finishes a minute later — the same result. “How?!” Adam points at +y and −y: “That pair of equations was asking to be added. When you see opposite signs, elimination is a shortcut.”

💡

Before solving, glance at the signs. Opposite pairs (+y and −y) = add at once. Same pairs (+y and +y) = first multiply one equation by −1 and then add.

The elimination method step by step

I will show it on: 2x + y = 11 and x − y = 1.

Step 1: Find what will cancel. By y there is +1 and −1 — opposite numbers. Great, I add straight away.

Step 2: Add the left sides and the right sides. Left: 2x + y + x − y = 3x. Right: 11 + 1 = 12. New equation: 3x = 12.

Step 3: Solve. x = 4.

Step 4: Find y by substituting into either original equation. I take the second: 4 − y = 1, so y = 3.

Check: 2 · 4 + 3 = 11 — it matches. 4 − 3 = 1 — it matches. Answer: [4; 3].

When nothing cancels by itself: the pair x + y = 7 and x + 2y = 10. By x there is +1 and +1 — same signs. I multiply the first equation by −1: −x − y = −7. Now I add with the second: −x − y + x + 2y = y and −7 + 10 = 3. You get y = 3, then x = 4.

You may multiply an equation by any number except zero — you just have to use it on both sides and every term.

Example 1: opposite signs, add at once

Question: Solve x + y = 12 and x − y = 4.

Method: By y the signs are opposite. I add the equations:

Left sides: x + y + x − y = 2x. Right sides: 12 + 4 = 16.

2x = 16, so x = 8.

I find y from the first equation: 8 + y = 12, so y = 4.

Check: 8 + 4 = 12 — it matches. 8 − 4 = 4 — it matches.

Answer: [8; 4].

Example 2: multiply first, then add

Question: Solve 2x + y = 14 and x + y = 9.

Method: By y there is +1 and +1 — adding would cancel nothing. I multiply the second equation by −1: −x − y = −9.

I add with the first: 2x + y − x − y = x and 14 − 9 = 5.

You get x = 5. I find y: 5 + y = 9, so y = 4.

Check: 2 · 5 + 4 = 14 — it matches. 5 + 4 = 9 — it matches.

Answer: [5; 4].

Example 3: multiplying by a bigger number

Question: Solve 3x + 2y = 19 and x − y = 3.

Method: I want to cancel y. In the first equation there is 2y, in the second −y. I multiply the second equation by two: 2x − 2y = 6. Now I have +2y and −2y.

I add: 3x + 2y + 2x − 2y = 5x and 19 + 6 = 25.

5x = 25, so x = 5. I find y from x − y = 3: 5 − y = 3, so y = 2.

Check: 3 · 5 + 2 · 2 = 15 + 4 = 19 — it matches. 5 − 2 = 3 — it matches.

Answer: [5; 2].

⚠️

If you multiply an equation, multiply EVERYTHING. A classic mistake: from x − y = 3 you make 2x − 2y = 3 and forget the right side. An equation is a scale — if you double the left pan and leave the right, the balance is gone and the result will be nonsense. Multiply every term on the left and the whole right side.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Pairs with a sum and a difference — made for the elimination method. Add the equations, y disappears. Answer with the value of x.

Practise the questions

On paper

Three pairs by elimination. Write the equations under each other, add by columns.

Solve x + y = 16 and x − y = 6. Add, find the other, check.
Solve 2x + y = 13 and x + y = 8. First multiply one equation by −1.
Solve 3x + 2y = 16 and x − 2y = 0. What cancels by itself?
Solve the last pair again by substitution as a check. It must come out the same.

Now you 💪

You can tell at a glance when something cancels by itself (opposite signs by the same unknown).
You can multiply a whole equation by a number — every term including the right side.
One pair came out the same by both methods.

✅

Done when: Three pairs solved by elimination, the checks match, and one pair came out the same by substitution too.

What to take from this lesson

Elimination method: add the equations so that one unknown cancels.
It works at once when an unknown has opposite signs (+y and −y).
If not, multiply one equation by a suitable number — the whole thing, both sides.
After x always find y from either original equation.
Both methods lead to the same result. Pick according to the question.