NCEA Level 1Simultaneous linear equationsAS 91947

Simultaneous equations: elimination and substitution

What this lesson covers

  • Two equations in two unknowns are solved by the pair of values that makes both true at once.
  • Graphically, that pair is the point where the two lines cross.
Key idea

One pair that fits both

  • x plus y is 10, and x minus y is 4.
  • Lots of pairs fit one equation. Only one fits both.
  • That pair, x is 7 and y is 3, is the solution.
Pairx + y = 10?x − y = 4?
x = 6, y = 4yes, 10no, 2
x = 8, y = 2yes, 10no, 6
x = 9, y = 5no, 14yes, 4
x = 7, y = 3yes, 10yes, 4
Key idea

Why you can add two equations

  • Each equation says two amounts are equal.
  • Add equal to equal and the totals are still equal.
  • Plus y and minus y cancel, so y is eliminated.
LineReasonResult
x + y = 10two equal amounts
x − y = 4two more equal amounts
Add left, add rightequal + equal is equal2x = 14
The y terms+y and −y cancelx = 7
Back in the first7 + y = 10y = 3
Worked example

Elimination by adding

  • Plus two y and minus two y: opposite signs.
  • Add, and eight x is thirty-two, so x is four.
  • y is three. Check it in both equations.
StepWorkingResult
3x + 2y = 185x − 2y = 14+2y and −2y
Add them8x = 32x = 4
x = 4 in the first12 + 2y = 18y = 3
Check the first12 + 6 = 18true
Check the second20 − 6 = 14true
Worked example

Elimination by subtracting

  • Both have plus two x: the same sign.
  • Take the second from the first. Two y is six.
  • y is three, x is two, and both checks hold.
StepWorkingResult
2x + 3y = 132x + y = 7both +2x
First take second2y = 6y = 3
y = 3 in the second2x + 3 = 7x = 2
Check the first4 + 9 = 13true
Check the second4 + 3 = 7true
Key idea

Add or subtract?

  • Look at one letter in both equations.
  • Opposite signs, add. Same signs, subtract.
  • If the sizes differ, multiply first.
CoefficientsDo thisBecause
Same size, opposite signaddthey cancel
Same size, same signsubtractthey cancel
Different sizesmultiply firstto make a match
Whichever it isone letter goesone unknown left
Worked example

Multiply one equation first

  • Three y and minus y do not match yet.
  • Times the whole second equation by three.
  • Now add. x is three, y is two, and both check.
StepWorkingResult
4x + 3y = 182x − y = 4no match yet
Second × 36x − 3y = 12every term
Add to the first10x = 30x = 3
x = 3 in the second6 − y = 4y = 2
Check both12 + 6 = 18, 6 − 2 = 4both true
Worked example

With an unknown constant

  • x plus y is k, where k is some fixed number.
  • Add for x, subtract for y. Nothing new.
  • The answers are in terms of k, and they check.
StepWorkingResult
x + y = kx − y = 6k is fixed
Add them2x = k + 6x = (k + 6)/2
Subtract them2y = k − 6y = (k − 6)/2
Check the sum2k/2k, true
Check the difference12/26, true
Worked example

Substitution

  • One equation already says what y is.
  • So put two x minus one in place of y.
  • x is three, y is five. Check in both.
StepWorkingResult
y = 2x − 13x + 2y = 19y is alone
Replace y3x + 2(2x − 1) = 19same number
Expand, collect7x − 2 = 19x = 3
Find y2 × 3 − 1y = 5
Check both5 = 5, 9 + 10 = 19both true
Key idea

The solution on a graph

  • Blue: y equals two x minus one.
  • Gold: three x plus two y equals nineteen.
  • Both sets of points pass through (3, 5).
The solution on a graph
Key idea

Why the crossing is the answer

  • Every point on a line fits its own equation.
  • Only the crossing point is on both lines.
  • Parallel lines never cross, so there is no solution.
PointLies onSo
A point on one linefits that equationonly
The crossingis on both linesfits both
(3, 5)same as the algebrax = 3, y = 5
Parallel linesnever crossno pair works
y = 2x + 1, y = 2x + 4subtract: 0 = 3false
Worked example

Forming the equations

  • Tickets to a school production: adults and students.
  • Say what each letter stands for first.
  • One equation counts people, one counts dollars.
FactEquationNote
Lettersa adult, s studenttickets sold
150 people camea + s = 150counts people
Adults $12students $5
$1,240 taken12a + 5s = 1240counts dollars
Worked example

Solving the ticket problem

  • Times the first by five so the s terms match.
  • Subtract: seven a is 490, so a is seventy.
  • s is eighty. Check people, then check money.
StepWorkingResult
First × 55a + 5s = 750s terms match
Second take that7a = 490a = 70
a = 70 in the first70 + s = 150s = 80
Check people70 + 80 = 150true
Check money840 + 400 = 1240true
Worked example

Answer the question that was asked

  • a is seventy is not yet an answer.
  • 70 adult tickets and 80 student tickets were sold.
  • Whole, positive numbers: that makes sense for tickets.
Summary

What to take away

  • Three ways in, one answer, and always two checks.
A solutionone pair that makes both equations true
Opposite signsadd the equations
Same signssubtract them
Nothing matchesmultiply first, and every term
y = ... alreadysubstitute it in
On a graphthe point where the two lines cross
Every timecheck in both original equations
Your turn

Your turn: four questions

  • Pause here and write all four out.
  • Question 3: define your letters first.
  • Check every answer in both equations.
Question
1Solve 3x + y = 11 and 2x − y = 4 by elimination.
2Solve y = x + 4 and 2x + 3y = 27 by substitution.
3Pies $5, sausage rolls $3: 50 sold for $196. How many of each?
4Where do the lines y = x + 1 and y = 7 − x cross?
Show answer

1. x = 3, y = 2. The y terms have opposite signs, so add: 5x = 15. Then 9 + y = 11. Check: 9 + 2 = 11 and 6 − 2 = 4. 2. x = 3, y = 7. Replace y: 2x + 3(x + 4) = 27, so 5x + 12 = 27 and 5x = 15. Check: 7 = 3 + 4 and 6 + 21 = 27. 3. 23 pies and 27 sausage rolls. p + r = 50 and 5p + 3r = 196; three times the first is 3p + 3r = 150, and subtracting gives 2p = 46. Check: 23 + 27 = 50 and 115 + 81 = 196. 4. At (3, 4). Both equal y, so x + 1 = 7 − x, 2x = 6, x = 3. Check: 3 + 1 = 4 and 7 − 3 = 4.

Worked example

The answers

  • x is three, y is two.
  • x is three, y is seven.
  • 23 pies and 27 sausage rolls.
  • They cross at (3, 4).

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