Simultaneous linear equations

Skills and methods in this topic, easy to hard

1. 1Solutions of an equation in two unknownsA pair that balances it is a solution; they all lie on one line

   Method(6, 1): 6 + 2 × 1 = 8, a solution

   Another way: y in terms of x, then a tabley = 8 − x2: x = 6, y = 1

2. 2Solving by graphsWhere the lines cross is the solution; check it in both

   Methodx + y = 5 and x − y = 1 meet at (3, 2)

   Another way: Check by substituting3 + 2 = 5, 3 − 2 = 1

3. 3Solving by substitutionWrite one unknown with the other; bracket it as you substitute

   Method3x + (2x − 1) = 14, x = 3, y = 5

   Another way: Elimination(2x − y) + (3x + y) = 1 + 14, 5x = 15

4. 4Solving by eliminationSame coefficient, subtract; opposite, add; multiply first to match

   Method(2x + 3y) + (4x − 3y) = 12 + 6, 6x = 18

   Another way: Substitutionx = y + 2 into 3x + 2y = 16

5. 5Simultaneous equations in other formsTidy into ax + by = c; 0 = 5, none; 0 = 0, endlessly many

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

   Another way: Substitute insteady = 6 − x, 2x − (6 − x) = 6, x = 4

6. 6From words to simultaneous equationsTwo unknowns need two different facts

   Method2x + 3y = 260, x − y = 30

   Another way: Just one unknown2(y + 30) + 3y = 260

7. 7Word problems with simultaneous equationsChoose, form, solve, then check it makes sense

   Methodx + y = 10, 2x + 4y = 32: x = 4, y = 6

   Another way: Suppose them all alikeAll chickens: 20 legs; (32 − 20) ÷ 2 = 6

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Solutions of an equation in two unknowns →

   In 3x + 3y = 9, when x = 4, y =

   ▶Show the answer and working

   Answer: −1

   Put in x = 4: 3 × (4) + 3y = 9, so 3y = −3 and y = −1.

2. Example 2Solving by elimination →

   4x + 3y = −15 and 2x + 3y = −9. How do you eliminate y?

   • A. Add the two
   • B. Double the first, then add
   • C. y can't be eliminated
   • D. Subtract one from the other

   Choose the right answer

   ▶Show the answer and working

   Answer: D. Subtract one from the other

   Same coefficient: subtract; opposite: add. Either way y is gone.

3. Example 3Word problems with simultaneous equations →

   A two-digit number's digits add to 12. Swapping its tens and ones digits gives a number 54 larger. What is the number?

   ▶Show the answer and working

   Answer: 39

   Tens digit x, ones digit y: x + y = 12, (10y + x) − (10x + y) = 54, so y − x = 6. Then x = 3, y = 9: the number is 39.