A pair of simultaneous equations · 跟马丁学编程

Two equations, two unknowns

One equation with one unknown — you can already do that. Today we step it up: two unknowns at once, x and y.

One equation is not enough for two unknowns. Take x + y = 10. x = 1 and y = 9 works. But also x = 4 and y = 6. Or x = 7.5 and y = 2.5. Infinitely many options — the equation itself does not decide which one holds.

That is why you need a second equation. When you add x − y = 4 to x + y = 10, suddenly only one pair fits: x = 7 and y = 3. Try another — one of the equations will always protest.

Two equations together are called simultaneous equations. The solution is not one number, but a pair of numbers — a value for x and a value for y that satisfy both equations at the same time.

Today you will learn to read a pair of equations, write it down and check a solution. Methods for finding the solution come in the next two lessons.

Jonáš gives Ema a riddle: “I am thinking of two numbers. Together they make 12.” Ema shrugs: “That could be 6 and 6, but also 1 and 11. Not enough information.” Jonáš adds: “And their difference is 2.” Ema scratches on paper for a bit: “Then it is 7 and 5. No other pair meets both conditions.” Jonáš nods: “That is exactly how simultaneous equations work. One sentence about two numbers is not enough. Two sentences — and the numbers are caught.”

💡

Always check a solution by substituting into BOTH equations. A pair that satisfies only one is not a solution.

How to read a pair of equations and check a solution

How you write it: two equations one under the other, often with a brace. For example:

x + y = 9 x − y = 3

What we look for: a pair of numbers x and y that satisfy both equations at once. You write it x = 6, y = 3, or as an ordered pair [6; 3] — the first number is always x, the second y.

How to check that a pair is a solution: substitute into both equations.

I try [6; 3]: first equation 6 + 3 = 9 — it matches. Second equation 6 − 3 = 3 — it matches. It is a solution.

I try [5; 4]: first equation 5 + 4 = 9 — it matches. Second equation 5 − 4 = 1, but it should be 3 — it does not match. It is not a solution, even though one equation passed.

A farm method for simple pairs: the “sum and difference” type you can solve by thinking. When two numbers have sum 9 and difference 3, the bigger number is (9 + 3)/2 = 6 and the smaller (9 − 3)/2 = 3. For harder pairs you will have proper methods from the next lesson. Today it is enough to understand what the pair of equations says and how you check.

Example 1: is [4; 2] a solution?

Question: Check whether the pair x = 4, y = 2 solves x + y = 6 and x − y = 2.

Method: Substitute into the first equation: 4 + 2 = 6. It should be 6. It matches.

Substitute into the second equation: 4 − 2 = 2. It should be 2. It matches.

Answer: Yes, [4; 2] is a solution — it satisfied both equations.

The pair [5; 1] would also satisfy the first equation (5 + 1 = 6), but not the second (5 − 1 = 4 ≠ 2). That is why you always test both.

Example 2: find the solution by thinking

Question: Two numbers have sum 14 and difference 4. Find them.

Method: Write it as simultaneous equations: x + y = 14, x − y = 4.

Thinking: if the numbers were the same, each would be 7. Difference 4 means one is 2 above seven and the other 2 below it.

So x = 9 and y = 5.

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

Answer: x = 9, y = 5, written [9; 5].

Example 3: simultaneous equations from a word problem

Question: Tereza bought a pen and a notebook for 42 Kč. The notebook was 8 Kč more expensive than the pen. Write the simultaneous equations (do not solve yet).

Method: Choose the unknowns: p = price of the pen, s = price of the notebook. Unknowns do not have to be called x and y.

First sentence: together 42 Kč, so p + s = 42.

Second sentence: the notebook 8 Kč more expensive, so s = p + 8.

Write: p + s = 42 and s = p + 8.

Two sentences from the question, two equations. Each sentence with a number usually gives one equation — that will help with word problems in lesson 7.

⚠️

One satisfied equation is not enough. The most common check mistake: you substitute into the first equation, it matches, and you call the pair a solution. But a solution of simultaneous equations must satisfy both equations at once. Always two substitution lines, not one.

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.

Simultaneous equations of the sum-and-difference type. Answer with the value of x — write a whole number.

Practise the questions

On paper

Reading, checking and one pair by thinking.

Check whether [5; 3] solves x + y = 8 and x − y = 2. Write both substitution lines.
Check whether [6; 2] solves x + y = 8 and x − y = 2. Write which equation failed.
By thinking find two numbers with sum 20 and difference 6. Do the check in both equations.
Write as simultaneous equations: two numbers have sum 30 and one is 10 bigger than the other. Do not solve, just write.

Now you 💪

You can explain why one equation is not enough for two unknowns.
You write a solution as a pair and check it in both equations.
You can solve a sum-and-difference pair by thinking and the check matches.

✅

Done when: You can tell whether a pair of numbers is a solution of simultaneous equations, and you have solved one simple pair by thinking with a check.

What to take from this lesson

Simultaneous equations = two equations that must hold at the same time.
The solution is a pair of numbers [x; y], not one number.
Checking = substituting into both equations. One is not enough.
Each sentence with a number in a word problem usually gives one equation.
Sum and difference: bigger number = (sum + difference)/2.