x + y = 3x − y = 4: x = □, y = □ Fill in each blank

2, 2. Split it into two: x + y = 4 and 3x − y = 4; adding removes y: x = 2, y = 2.

3(x + 5) + y = 5, x + 3(y − 1) = −9: x = □, y = □ Fill in each blank

−3, −1. Open the brackets and tidy into ax + by = c, then eliminate: x = −3, y = −1.

Multiplying both sides of x/3 + y/6 = 2 by 18 gives 6x + 3y = 2. True or false?

False. Every term is multiplied by 18, the 2 on the right too: 6x + 3y = 18 × 2.

Simultaneous linear equations · Ages 13–14

Simultaneous equations in other forms

Tidy into ax + by = c; 0 = 5, none; 0 = 0, endlessly many.

How to do it

Method

Tidy into ax + by = c; 0 = 5, none; 0 = 0, endlessly many

x + y = 2x − y = 6: x + y = 6, 2x − y = 6

Another way

Substitute instead

y = 6 − x, 2x − (6 − x) = 6, x = 4

In full

Tidy each equation into the shape ax + by = c first, then choose a method: clear fractions by multiplying by the L.C.M. of the denominators, open brackets. x + y = 2x − y = 6 is really two equations, x + y = 6 and 2x − y = 6; adding them, 3x = 12, x = 4, y = 2. Which way is quicker? A coefficient of 1: substitute. The same or opposite coefficients: add or subtract. If eliminating leaves something false like 0 = 5, there is no solution (the lines are parallel); if it leaves 0 = 0, there are endlessly many (the lines lie on each other).

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
x + y = 3x − y = 4: x = , y =
Fill in each blank
Show the answer and working
Answer: 2, 2
Split it into two: x + y = 4 and 3x − y = 4; adding removes y: x = 2, y = 2.
Example 2
3(x + 5) + y = 5, x + 3(y − 1) = −9: x = , y =
Fill in each blank
Show the answer and working
Answer: −3, −1
Open the brackets and tidy into ax + by = c, then eliminate: x = −3, y = −1.
Example 3
Multiplying both sides of x3 + y6 = 2 by 18 gives 6x + 3y = 2.
True or false?
Show the answer and working
Answer: False
Every term is multiplied by 18, the 2 on the right too: 6x + 3y = 18 × 2.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.How many solutions do 3x + 4y = 12 and 9x + 12y = 36 have together?
- A. Exactly one
- B. Endlessly many
- C. Two
- D. None
Choose the right answer
Answer:
2.3x + y = 5x − y = −8: x = , y =
Fill in each blank
Answer:
3.Multiplying both sides of x6 + y2 = 1 by 12 gives 2x + 6y = 12.
True or false?
Answer:
4.2(x + 1) + y = 7, x + 2(y − 1) = 5: x = , y =
Fill in each blank
Answer:
5.x + y2 = x − y3 = 2: x = , y =
Fill in each blank
Answer:

All the answers

1. B. Endlessly many
2. −2, −2
3. True
4. 1, 3
5. 5, −1

Free worksheet: Simultaneous equations in other formsWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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