Algebra · Junior Cert

Simultaneous Equations

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Section 1 of 4

The Elimination Method

Two equations, two unknowns. Make the numbers in front of one letter match, then add or subtract to make that letter disappear.

Worked example — just add

Solve $x+y = 5$ and $x-y = 1$.

Add the two: $2x = 6 \Rightarrow x = 3$

Into $x+y=5$: $3+y = 5$

$x = 3,\ y = 2$

Worked example — match first

Solve $2x+3y = 7$ and $5x+2y = 12$.

$\times 2$ and $\times 3$: $\ 4x+6y = 14,\quad 15x+6y = 36$

Subtract: $11x = 22 \Rightarrow x = 2$

Into the first: $4+3y = 7 \Rightarrow 3y = 3$

$x = 2,\ y = 1$

Worked example

Solve $3x-5y = 8$ and $2x-7y = 9$.

$\times 7$ and $\times 5$: $\ 21x-35y = 56,\quad 10x-35y = 45$

Subtract: $11x = 11 \Rightarrow x = 1$

$3-5y = 8 \Rightarrow -5y = 5$

$x = 1,\ y = -1$

Worked example — fraction answers are fine

Solve $5x-y = 7$ and $x+8y = 3$.

$\times 8$: $\ 40x-8y = 56$; add $x+8y = 3$

$41x = 59 \Rightarrow x = \dfrac{59}{41}$

$x = \dfrac{59}{41},\ y = \dfrac{8}{41}$

You try

Solve $3x-8y = 3$ and $x-7y = 4$.

Pen and paper out — try it before you reveal.

$\times 1,\times 3$: $3x-8y=3,\ 3x-21y=12$

Subtract: $13y = -9 \Rightarrow y = -\tfrac{9}{13}$

$x-7(-\tfrac{9}{13}) = 4$

$x = -\dfrac{11}{13},\ y = -\dfrac{9}{13}$

Section 2 of 4

When There’s a Fraction

Clear the fractions first — multiply the equation across — then eliminate as usual.

Worked example

Solve $5x+6y = 16$ and $\dfrac{x}{2}+y = 2$.

Multiply the second by $2$: $x+2y = 4$

$\times 3$: $3x+6y = 12$; subtract from $5x+6y=16$: $2x = 4$

$x = 2,\ y = 1$

Worked example

Solve $\dfrac{x}{3}+\dfrac{y}{5} = 3$ and $2x+y = 16$.

First $\times 15$: $5x+3y = 45$

$2x+y=16$, $\times 3$: $6x+3y = 48$; subtract: $-x = -3 \Rightarrow x = 3$

$x = 3,\ y = 10$

You try

Solve $\dfrac{1}{3}x+\dfrac{1}{4}y = 2$ and $2x+5y = 26$.

Pen and paper out — try it before you reveal.

First $\times 12$: $4x+3y = 24$

With $2x+5y=26$ ($\times 2$): $4x+10y=52$; subtract: $-7y=-28 \Rightarrow y=4$

$4x+12=24$

$x = 3,\ y = 4$

Section 3 of 4

The Graphical Method

Each equation is a line. Draw both — where they cross is the solution.

Worked example

Draw $x+y = 3$ and $x-y = 1$ and state the point of intersection.

$x+y=3$: points $(0,3)$ and $(3,0)$

$x-y=1$: points $(0,-1)$ and $(1,0)$

They cross at $(2,1)$, so $x=2,\ y=1$.

Section 4 of 4

Worded Problems

Name your two unknowns with letters, turn each sentence into an equation, then solve.

Worked example

Two apples and a pear cost $50$c. Three apples and two pears cost $80$c. Find the cost of one apple and one pear.

$2a+p = 50,\quad 3a+2p = 80$

$\times 2$: $4a+2p = 100$; subtract $3a+2p=80$: $a = 20$

$40+p = 50$

Apple $= 20$c, pear $= 10$c

Worked example

Five hurls and three balls cost €$115$. Four hurls and five balls cost €$105$. Find the cost of one hurl and one ball.

$5h+3b = 115,\quad 4h+5b = 105$

$\times 5,\times 3$: $25h+15b = 575,\ 12h+15b = 315$; subtract: $13h = 260 \Rightarrow h = 20$

$100+3b = 115 \Rightarrow 3b = 15$

Hurl $=$ €$20$, ball $=$ €$5$

Worked example

A farmer has pigs and hens. There are $11$ heads and $34$ legs. How many of each?

$p+h = 11$ (heads), $4p+2h = 34$ (legs)

$\div 2$: $2p+h = 17$; subtract $p+h=11$: $p = 6$

$6$ pigs, $5$ hens

Worked example

The sum of two numbers is $32$. Their difference is $8$. Find the numbers.

$x+y = 32,\quad x-y = 8$

Add: $2x = 40 \Rightarrow x = 20$; $20+y=32$

$20$ and $12$

You try

An adult and $2$ children cost €$20$ at the cinema. Two adults and $3$ children cost €$35$. Find the cost for one adult and one child.

Pen and paper out — try it before you reveal.

$x+2y = 20,\quad 2x+3y = 35$

$\times 2$: $2x+4y=40$; subtract $2x+3y=35$: $y = 5$

$x+10=20$

Adult $=$ €$10$, child $=$ €$5$

You try

Three cars and $2$ buses occupy $110\text{ m}^2$. Four cars and $3$ buses occupy $160\text{ m}^2$. Find the space for one car and one bus.

Pen and paper out — try it before you reveal.

$3x+2y = 110,\quad 4x+3y = 160$

$\times 3,\times 2$: $9x+6y=330,\ 8x+6y=320$; subtract: $x=10$

$30+2y=110$

Car $= 10\text{ m}^2$, bus $= 40\text{ m}^2$

That’s simultaneous equations.

Match a letter and eliminate, clear any fractions first, or read them off a graph — and always turn words into two equations.