A pair of simultaneous equations · Lập trình với Martin

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. Bắt đầu 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.