The substitution method · 跟马丁学编程

Make it the subject and substitute

Last time you cracked simultaneous equations by thinking. That works with nice numbers, but you need a method that can handle anything. The first one is called substitution.

The idea is simple: two unknowns are too many, so I write one in terms of the other and that cancels it.

When the first equation says y = x + 1, I know everything about y — it is just “x plus one”. So I substitute that for y into the second equation. Suddenly there is only x in it. And you already know how to solve an equation with one unknown.

Then you go back: you put the calculated x into y = x + 1 and you have y too. Two unknowns, two steps.

Substitution is best when one unknown is already the subject — when you see “y = something” somewhere, or you can make it so in one move. In entrance exams it is the most used method. Come and train it.

Sofie is solving y = x + 2 and x + y = 10. She stares at it and then laughs: “Wait, the first equation tells me straight away what y is! So I just write that in.” She rewrites the second equation: x + (x + 2) = 10. “Now it is an ordinary equation. 2x + 2 = 10, so x = 4. And y is x plus two, so 6.” Filip looks over her shoulder: “That is all? No magic?” No magic. Just swapping a letter for what it equals.

💡

Always wrap the expression you substitute in a bracket: instead of y write (x + 2). The bracket protects you when a multiply or a minus appears in front of y.

The substitution method in four steps

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

Step 1: Pick the equation where an unknown is easy to make the subject. Here it is 完成 — the second equation says y = x + 2. If it were not, you make it yourself: from x + y = 7 you get y = 7 − x in one move.

Step 2: Substitute into the other equation. Into 2x + y = 11 I put the bracket (x + 2) instead of y: 2x + (x + 2) = 11.

Step 3: Solve the equation with one unknown. 2x + x + 2 = 11, so 3x + 2 = 11, so 3x = 9, so x = 3.

Step 4: Find the other unknown. I substitute x = 3 into y = x + 2: y = 5.

Check (in both equations, of course): 2 · 3 + 5 = 11 — it matches. 5 = 3 + 2 — it matches.

Answer: x = 3, y = 5, so [3; 5].

The whole method stands on one move: you replace a letter with the expression it equals. There is nothing more in it.

Example 1: y is already the subject

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

Method: The second equation says y is twice x. Substitute into the first: x + 2x = 9.

Add: 3x = 9, so x = 3.

Find y: y = 2 · 3 = 6.

Check: 3 + 6 = 9 — it matches. 6 = 2 · 3 — it matches.

Answer: [3; 6].

Example 2: make it the subject first, then substitute

Question: Solve x + y = 8 and 3x + 2y = 19.

Method: From the first equation I make y the subject: y = 8 − x.

I substitute into the second, with a bracket: 3x + 2 · (8 − x) = 19.

Expand: 3x + 16 − 2x = 19. Combine: x + 16 = 19, so x = 3.

Find y: y = 8 − 3 = 5.

Check: 3 + 5 = 8 — it matches. 3 · 3 + 2 · 5 = 9 + 10 = 19 — it matches.

Answer: [3; 5].

Example 3: why the bracket is necessary

Question: Solve y = x − 1 and 10 − 2y = 3x.

Method: I substitute y into the second equation. In front of y there is “minus two times”, so the bracket is vital: 10 − 2 · (x − 1) = 3x.

I expand including the sign: 10 − 2x + 2 = 3x. Minus two times minus one is plus two.

I combine: 12 − 2x = 3x, so 12 = 5x, so x = 2.4.

I find y: y = 2.4 − 1 = 1.4.

Check: 10 − 2 · 1.4 = 10 − 2.8 = 7.2 and 3 · 2.4 = 7.2 — it matches.

Answer: [2.4; 1.4]. Without the bracket a completely different number would come out.

⚠️

Substituting without a bracket is a classic error. When you put y = x − 1 into 10 − 2y as 10 − 2x − 1, you lost the sign: correctly it is 10 − 2 · (x − 1) = 10 − 2x + 2. A minus in front of a bracket changes the signs of everything inside. Always write the bracket, even if it seems pointless — it breaks nothing and it will not let a mistake through.

Practice: now you do it 🎲

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

Train simultaneous equations. Feel free to solve them by substituting — from the second equation make x the subject and put it into the first. Answer with the value of x.

Practise the questions

On paper

Three pairs by the substitution method. Each time: make it the subject, substitute with a bracket, solve, find the other, check.

Solve y = x + 3 and x + y = 11. Check the result in both equations.
Solve x + y = 10 and 2x + y = 14. First make y the subject from the first equation.
Solve y = 2x and 3x − y = 4.
For one pair, try substituting without a bracket on purpose and compare what comes out. You will see why the bracket is written.

Now you 💪

From the equation x + y = 7 you can make y the subject in one move.
You substitute with a bracket and watch the signs when you expand.
Every pair has a check in both equations.

✅

Done when: You have three pairs solved by substitution, all the checks match and you know what the bracket is for when you substitute.

What to take from this lesson

Substitution method: make one unknown the subject and put it into the other equation.
Always put the substituted expression in a bracket.
After solving x, do not forget to find y — the answer is a pair.
The method is useful when one unknown is already the subject or you can make it so in one move.
A check in both equations closes every question.