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Simultaneous

Simultaneous

Simultaneous equations by elimination

Elimination Core + Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Two equations, one pair of answers
  4. Forming them from words
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 2.5: simultaneous linear equations
  • 2.5Solve two simultaneous linear equations by elimination.
  • 2.5Multiply one or both equations first so a letter matches.
  • 2.5Form simultaneous equations from a problem and answer in context.

Before you start: Linear equations. Next: Substitution, and a line with a curve.

Formula bank

Elimination ruleLearn itsame signs → subtract; different signs → addapplies to the letter you want to remove, once its numbers match.Use it when two linear equations, letters lined up.

Two equations, one pair of answers

\(x+y=10\) has endless solutions; so does \(x-y=4\). Simultaneous equations want the ONE pair \((x,y)\) that makes both true - on a graph, where the two lines cross. Adding or subtracting the equations keeps them balanced and can make one letter disappear.

General methodHow to solve by elimination

  1. Line the equations up (x under x, y under y, numbers on the right) and label them (1) and (2).
  2. Make the numbers in front of one letter the same: multiply one or both WHOLE equations.
  3. Same signs: subtract. Different signs: add. Solve for the letter that is left.
  4. Substitute that value into either original equation to find the other letter.
  5. Check both values in the other equation.

Example Solve \(4x+3y=10\) and \(2x-5y=18\).

Multiply (2) by 2 to get \(4x\) in both, then subtract.

  1. (2) × 2: \(4x-10y=36\)
  2. (1) − new: \(13y=-26\Rightarrow y=-2\)
  3. \(4x-6=10\Rightarrow x=4\)
  4. check (2): \(8+10=18\) ✓

Answer \(x=4,\ y=-2\)

Try it yourself

1
Solve \(x+y=10\) and \(x-y=4\).
Show answer

Answer \(x=7,\ y=3\)

2
Solve \(2x+y=11\) and \(x+y=7\).
Show answer

Answer \(x=4,\ y=3\)

3
Solve \(3x+4y=5\) and \(2x-3y=9\).
Hint
× (1) by 2 and (2) by 3 to get \(6x\).
Show answer

Answer \(x=3,\ y=-1\)

Forming them from words

General methodHow to set up simultaneous equations from a problem

  1. Choose two letters for the two unknowns (say what each means).
  2. Write one equation for each fact given.
  3. Solve by elimination and answer the question with units.

Example 2 adult tickets and 3 child tickets cost $270. 1 adult and 2 child tickets cost $160. Find the cost of each ticket.

a = adult, c = child: two equations, then eliminate.

  1. \(2a+3c=270\), \(a+2c=160\)
  2. (2) × 2: \(2a+4c=320\)
  3. subtract: \(c=50\)
  4. \(a=160-100=60\)

Answer adult $60, child $50

Ways they test it

Type 1 Solve, showing working

"Solve \(5x-2y=19\) and \(3x+4y=1\). You must show all your working." (3 marks)

Spot it two linear equations.

  1. Match one letter (here: (1) × 2 gives −4y).
  2. Add or subtract.
  3. Substitute back.

For example \(10x-4y=38\) and \(3x+4y=1\)

  1. add: \(13x=39\Rightarrow x=3\)
  2. \(9+4y=1\Rightarrow y=-2\)

Answer \(x=3,\ y=-2\)

Type 2 Decimals

"Solve \(2x+3y=3.4\) and \(5x+2y=4.1\)."

Spot it decimal right-hand sides: same method.

  1. Match a letter.
  2. Work carefully with the decimals.

For example (1)×2, (2)×3: \(4x+6y=6.8\), \(15x+6y=12.3\)

  1. \(11x=5.5\Rightarrow x=0.5\)

Answer \(x=0.5,\ y=0.8\)

Type 3 In context

"4 cups and 5 plates weigh 105 g ... 2 cups and 3 plates weigh 59 g. Find the mass of a cup." (4 marks)

Spot it two items, two totals.

  1. Two letters, two equations.
  2. Eliminate.
  3. Answer the question asked.

Mistakes to avoid

  • ✗ \(2(x-5y=18)\) as \(2x-10y=18\)✓ \(4x-10y=36\)Multiply EVERY term, including the right-hand side.
  • ✗ \(4x+3y\) minus \(4x-10y\) = \(-7y\)✓ \(13y\)\(3y-(-10y)=13y\).
  • ✗ Answers with no working✓ show each stepWorking is required for the marks.

Summary

  • Match the numbers in front of one letter (multiply whole equations).
  • Same signs subtract, different signs add.
  • Substitute back; check in the other equation.
  • The answer is one pair: where the two lines cross.

If you see …

  • "show all your working"write the multiplied equations and the add/subtract step.
  • two items and two totalstwo letters, two equations.

Practice

Practice: easy, medium, hard

Easy

1
Solve \(3x+2y=16\) and \(x-2y=0\).
Hint
Different signs: add.
Show answer

Answer \(x=4,\ y=2\)

2
Solve \(5x+3y=21\) and \(2x+3y=12\).
Hint
Same signs: subtract.
Show answer

Answer \(x=3,\ y=2\)

3
Solve \(3x-2y=21\) and \(5x+2y=51\).
Show answer

Answer \(x=9,\ y=3\)

Medium

1
Solve \(x+4y=9\) and \(3x-2y=13\).
Guide me

What is it asking? x and y

What do we know? × (2) by 2 gives −4y

Plan then add

Show working
  1. \(6x-4y=26\)
  2. add: \(7x=35\Rightarrow x=5\)
  3. \(5+4y=9\Rightarrow y=1\)

Answer \(x=5,\ y=1\)

2
Solve \(5x-2y=19\) and \(3x+4y=1\).
Guide me

What is it asking? x and y

What do we know? × (1) by 2 gives −4y

Plan then add

Show working
  1. \(10x-4y=38\)
  2. \(13x=39\Rightarrow x=3\)
  3. \(9+4y=1\Rightarrow y=-2\)

Answer \(x=3,\ y=-2\)

3
2 adult and 3 child tickets cost $270; 1 adult and 2 child tickets cost $160. Find the cost of a child ticket.
Guide me

What is it asking? child ticket

What do we know? \(2a+3c=270\), \(a+2c=160\)

Plan × (2) by 2, subtract

Show working
  1. \(2a+4c=320\)
  2. \(c=50\)

Answer $50

Hard

1
Solve \(2x+3y=3.4\) and \(5x+2y=4.1\).
Guide me

What is it asking? x and y

What do we know? match y: (1) × 2, (2) × 3

Plan subtract

Show working
  1. \(4x+6y=6.8\)
  2. \(15x+6y=12.3\)
  3. \(11x=5.5\Rightarrow x=0.5\)
  4. \(1+3y=3.4\Rightarrow y=0.8\)

Answer \(x=0.5,\ y=0.8\)

PartAnswer / workingMarksGuidance
coefficients matchedM1
one letter eliminatedM1
\(x=0.5,\ y=0.8\)A1
Total: 3 marks
2
Solve \(3x+4y=5\) and \(2x-3y=9\).
Guide me

What is it asking? x and y

What do we know? match x: × 2 and × 3

Plan \(6x+8y=10\), \(6x-9y=27\)

Show working
  1. subtract: \(17y=-17\Rightarrow y=-1\)
  2. \(3x-4=5\Rightarrow x=3\)

Answer \(x=3,\ y=-1\)

PartAnswer / workingMarksGuidance
\(6x+8y=10\) and \(6x-9y=27\)M1
\(17y=-17\)M1
x = 3, y = −1A1
Total: 3 marks
3
4 cups and 5 plates weigh 105 g. 2 cups and 3 plates weigh 59 g. Find the mass of a cup.
Guide me

What is it asking? a cup

What do we know? \(4c+5p=105\), \(2c+3p=59\)

Plan × (2) by 2, subtract

Show working
  1. \(4c+6p=118\)
  2. subtract: \(p=13\)
  3. \(2c+39=59\Rightarrow c=10\)

Answer 10 g

PartAnswer / workingMarksGuidance
both equationsB1
eliminationM1
p = 13A1
c = 10 gA1
Total: 4 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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