M4.8.5 Chapter 5

Two equations: adding or taking away

I know x + y = 10 and x − y = 2. How do I find both numbers?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

A x = 5 and y = 5, because 5 + 5 = 10.
You picked A. Read on. At the end, the chapter comes back to your guess.
B x = 6 and y = 4, because 6 + 4 = 10 and 6 − 4 = 2.
You picked B. Read on. At the end, the chapter comes back to your guess.
C x = 8 and y = 2, because 8 + 2 = 10.
You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

Solve two equations by getting rid of one letter
Multiply an equation first so the numbers match

Amina thinks of two numbers, x and y. She says: "Their sum is 10." That alone has many answers. 4 and 6 work, and so do 5 and 5. One clue is not enough for two unknown numbers.

Every pair adds to 10. Only x = 6 and y = 4 also gives x − y = 2.

Amina adds a second clue: "Their difference is 2", so x − y = 2. Now both equations must be true at the same time. Two equations like this are called simultaneous equations.

Look at the y terms. One equation has +y and the other has −y. If you add the two equations, the y terms cancel. This is elimination: you remove one letter.

Why may you add two equations? x + y and 10 are the same amount, and so are x − y and 2. Add equal amounts to equal amounts and the totals are equal too. It is like putting the same weight on both pans of a level balance. Taking equal amounts from equal amounts works the same way.

Draw it first

Add the equations

1 Write the two equations one under the other, with the same letters in the same columns.
2 The y terms are +y and −y. They cancel when you add, so mark them.
3 Add the left sides and add the right sides. x + x = 2x, and 10 + 2 = 12. So 2x = 12.

Worked example

Solve x + y = 10 and x − y = 2.

Add the equations. The y terms cancel: 2x = 12.
Divide both sides by 2: x = 6.
Put x = 6 in x + y = 10. Then 6 + y = 10, so y = 4.
Check in the other equation, x − y = 2: 6 − 4 = 2.

Answer: x = 6, y = 4

Sometimes both equations have the same term. Then the sign of the terms is the same, so you subtract. Take 3x + 2y = 19 and x + 2y = 9. Both have +2y.

The 2y terms match and have the same sign, so subtract: 3x − x = 2x and 19 − 9 = 10.

Worked example

Solve 3x + 2y = 19 and x + 2y = 9.

The y terms are both +2y, so subtract equation (2) from equation (1). Then 2x = 10.
Divide both sides by 2: x = 5.
Put x = 5 in equation (2): 5 + 2y = 9, so 2y = 4 and y = 2.
Check in equation (1): 3 × 5 + 2 × 2 = 19.

Answer: x = 5, y = 2

The rule is: if the signs of the matching terms are different, add. If they are the same, subtract. Always do the same to every term, on both sides of the equals sign.

What if no letter matches? Multiply an equation first. Multiplying every term of an equation by the same number keeps it true. Use it to make one letter match.

Multiply (1) by 3 and (2) by 2. Then both have 6x.

Call the new equations (3) and (4). The 6x terms match and have the same sign. Subtract: 9y − 4y = 5y and 36 − 26 = 10.

Worked example

Solve 2x + 3y = 12 and 3x + 2y = 13.

Make the x terms match. Multiply (1) by 3 and (2) by 2. Call the new equations (3) and (4): 6x + 9y = 36 and 6x + 4y = 26.
The x terms have the same sign, so subtract (4) from (3): 5y = 10.
Divide both sides by 5: y = 2.
Put y = 2 in equation (1): 2x + 6 = 12, so 2x = 6 and x = 3.
Check in equation (2): 3 × 3 + 2 × 2 = 13.

Answer: x = 3, y = 2

See it three ways

With things

Two bags of rice and one bag of beans weigh 7 kg. One bag of rice and one bag of beans weigh 4 kg. How heavy is a bag of rice?

As a picture

In words

The first weighing has one more bag of rice. The extra 7 − 4 = 3 kg is one bag of rice. The beans are 4 − 3 = 1 kg.

In symbols

2r + b = 7 and r + b = 4. Subtract: r = 3. Put it back: 3 + b = 4, so b = 1.

Try it

The sum and the difference

You need: paper and a pencil, a friend.

Think of two whole numbers. Say them to yourself and keep them secret.
Tell your friend their sum and their difference, the bigger minus the smaller.
Your friend adds the sum and the difference, then halves the answer. That is your bigger number.

What you will see: The sum plus the difference is twice the bigger number, because the smaller number cancels. Take the bigger number from the sum to find the smaller.

If you cannot try it, this is what happens.

A wrong idea many people have

To eliminate a letter you always add the equations.

Why it feels right: The first example was solved by adding, so adding feels like the method.

The test that breaks it: Take x + y = 10 and x + 3y = 14. Adding gives 2x + 4y = 24, which still has both letters. Subtracting gives −2y = −4, so y = 2, which has one letter.

What is true: Add when the matching terms have different signs. Subtract when they have the same sign.

A wrong idea many people have

When you multiply an equation, you multiply only the term you want to match.

Why it feels right: You are only trying to change one number, so changing one number feels enough.

The test that breaks it: Multiply x + y = 4 by 3 on the x only: 3x + y = 4. Now x = 1, y = 3 does not fit it: 3 + 3 = 6, not 4. But 1 + 3 = 4 fitted the original.

What is true: Multiply every term on both sides by the same number. Then the new equation says the same thing as the old one.

Key words

simultaneous equations: two or more equations that must be true at the same time.
elimination: removing one letter by adding or subtracting the equations.

Back to your guess

The answer is B: x = 6 and y = 4, because 6 + 4 = 10 and 6 − 4 = 2.

Both equations must be true at once. x = 6 and y = 4 fit both: 6 + 4 = 10 and 6 − 4 = 2. The other pairs fit only the first equation.

Keep it

Copy this into your exercise book.

Match one letter, multiplying an equation if you must. Different signs: add. Same signs: subtract. Find the other letter, then check in both equations.

Draw: Two equations under each other with the matching column hatched, a line, and the result row under it.