Simultaneous Equations flashcards
9 cards on Simultaneous Equations: the key points, worked examples and the slips to watch. Swipe through them, or print them.
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Simultaneous Equations
What simultaneous equations are
What are simultaneous equations, and what does solving them mean?
Answer it in your head, then turn the card over.
What simultaneous equations are
Key points
- Two equations with the same unknowns, solved together.
- The solution is the pair of values that makes both equations true at once.
- One equation on its own has endless solutions: the second one pins down a single pair.
x + y = 10 and x − y = 2
x + y = 10 fits 1 and 9, 5 and 5, 6 and 4, and many more.
Only x = 6, y = 4 fits x − y = 2 as well: 6 + 4 = 10 and 6 − 4 = 2.
Watch out
A solution is always a pair: x = 6 on its own is only half the answer.
Simultaneous Equations
Elimination
How do you solve simultaneous equations by elimination when one letter's coefficients already match?
Answer it in your head, then turn the card over.
Elimination
Key points
- Line the equations up: x under x, y under y, numbers on the right.
- Same signs, subtract the equations; opposite signs, add them.
- That removes one letter: solve for the other, then substitute back.
3x + 5y = 19 and 3x − 2y = 5
Both have +3x, the same sign, so subtract.
5y − (−2y) = 7y and 19 − 5 = 14, so 7y = 14 and y = 2
3x + 10 = 19, so 3x = 9 and x = 3
Watch out
Taking away a negative adds: 5y − (−2y) is 7y, not 3y. Put the subtraction in brackets when signs are mixed.
Simultaneous Equations
Matching the coefficients first
What do you do when neither letter has matching coefficients in the two equations?
Answer it in your head, then turn the card over.
Matching the coefficients first
Key points
- Multiply one equation, or both, so one letter has the same coefficient in each.
- Multiply every term on both sides, the number on the right included.
- Choose the letter that needs the smallest multipliers, then eliminate as usual.
2x + 3y = 13 and 4x − y = 5
Multiply 4x − y = 5 by 3: 12x − 3y = 15
Add it to 2x + 3y = 13: 14x = 28, so x = 2
4 + 3y = 13, so 3y = 9 and y = 3
Watch out
Scale the right-hand side too: 3 × (2x + y = 7) is 6x + 3y = 21, not 6x + 3y = 7.
Simultaneous Equations
Finding the second letter and checking
Once you have found one unknown, how do you finish the question and check your answer?
Answer it in your head, then turn the card over.
Finding the second letter and checking
Key points
- Substitute the value into either original equation and solve for the other letter.
- Test both values in the equation you did not use: it should come out true.
- Give both values, or the pair as a coordinate if the question asks for one.
3x + 2y = 12 and x + 2y = 8
Subtracting gives 2x = 4, so x = 2
In 3x + 2y = 12: 6 + 2y = 12, so y = 3
Check in x + 2y = 8: 2 + 6 = 8, which is true.
Watch out
Stopping after x loses marks: a question that says solve wants both values.
Simultaneous Equations
Substitution
How do you solve simultaneous equations by substitution, and when is it the best method?
Answer it in your head, then turn the card over.
Substitution
Key points
- Rearrange one equation to y = … (or x = …), unless it is already like that.
- Replace that letter in the other equation, in brackets, and solve.
- Put the value back into the rearranged equation to find the other letter.
- Best when one equation already has a letter on its own.
y = 3x − 2 and 2x + y = 13
2x + (3x − 2) = 13, so 5x − 2 = 13 and 5x = 15
x = 3, and y = 3 × 3 − 2 = 7
Check: 2 × 3 + 7 = 13
Watch out
Bracket the whole expression: 3(4 − x) is 12 − 3x, not 12 − x.
Simultaneous Equations
Solving with graphs
How do graphs solve simultaneous equations, and what does it mean when the lines are parallel?
Answer it in your head, then turn the card over.
Solving with graphs
Key points
- Each equation is a line: plot both on the same axes.
- The solution is where the lines cross, the only point that lies on both.
- Read the crossing point as (x, y) and check it in both equations.
- Parallel lines never cross, so the equations have no solution.
y = x + 1 and y = 5 − x
The lines cross at (2, 3), so x = 2 and y = 3.
Check: 2 + 1 = 3 and 5 − 2 = 3
Watch out
A point is written (x, y): a crossing at (3, 1) means x = 3 and y = 1, not the other way round.
Simultaneous Equations
Worded problems
How do you turn a worded problem with two unknowns into simultaneous equations and solve it?
Answer it in your head, then turn the card over.
Worded problems
Key points
- Say what each letter stands for, with its units: let a be an adult ticket in £.
- Two unknowns need two facts: write one equation from each.
- Solve, then answer the question asked, in words and units.
Ticket prices
4 adult and 3 child tickets cost £47; 2 adult and 3 child cost £31.
4a + 3c = 47 and 2a + 3c = 31
Subtract: 2a = 16, so a = 8. Then 16 + 3c = 31, so c = 5.
An adult ticket is £8 and a child ticket is £5.
Watch out
"An adult ticket costs £4 more than a child ticket" is a = c + 4, not c = a + 4.
Simultaneous Equations
HigherOne linear and one quadratic equation
How do you solve a linear and a quadratic equation simultaneously?
Answer it in your head, then turn the card over.
One linear and one quadratic equation
Key points
- Rearrange the linear equation to y = … (or x = …).
- Substitute it into the quadratic and rearrange to … = 0.
- Solve the quadratic: there are usually two x values.
- Put each x into the linear equation to find its own y, and pair them up.
y = x + 1 and y = x² − 5
x² − 5 = x + 1, so x² − x − 6 = 0
(x − 3)(x + 2) = 0, so x = 3 or x = −2
y = x + 1 gives y = 4 or y = −1: the points (3, 4) and (−2, −1)
Watch out
Keep each x with its own y: the answers are (3, 4) and (−2, −1), never (3, −1).
Simultaneous Equations
HigherA line and a circle
How do you find where a straight line meets a circle such as x² + y² = 13?
Answer it in your head, then turn the card over.
A line and a circle
Key points
- Substitute the line's equation into the circle's, expanding the bracket squared in full.
- Solve the quadratic: two answers mean two crossing points, one means the line touches, none means they never meet.
- Find each y from the line, not the circle: the circle gives two y values for one x.
y = x − 1 and x² + y² = 13
x² + (x − 1)² = 13, so 2x² − 2x + 1 = 13
2x² − 2x − 12 = 0, so x² − x − 6 = 0
(x − 3)(x + 2) = 0, so x = 3 or x = −2
y = x − 1 gives (3, 2) and (−2, −3)
Watch out
(x − 1)² is x² − 2x + 1, not x² + 1: write it as (x − 1)(x − 1) and expand.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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