Solving Simultaneous Equations

01Solving Simultaneous Equations

Elimination

This section covers eliminating an unknown from two linear equations and checking the resulting solution pair.

Remove one unknown

Simultaneous equations share unknowns whose values must satisfy all the equations together. A solution in two unknowns is a pair $(x,y)$ , not a value of just one variable.

A linear equation in two unknowns represents a straight line. Elimination removes one unknown by adding or subtracting suitable multiples of the equations.

Match the coefficients

Choose the unknown to remove. Multiply one or both equations so its coefficients have equal magnitude, multiplying every term on both sides.
Add or subtract the equations as shown below, then solve the equation in the remaining unknown.
Substitute that value into an original equation to find its partner, then check the pair in the other original equation.

Coefficients of the chosen unknown	Operation
Equal, with the same sign	Subtract one equation from the other.
Equal magnitude, with opposite signs	Add the equations.

02Solving Simultaneous Equations

Substitution

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Replace an unknown with an expression

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1Make a subject

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2Substitute

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3Find its partner

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03Solving Simultaneous Equations

Linear & Quadratic Equations

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Reduce to one unknown

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Worked example

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$2x-3y=23,\qquad 3y^2=4x^2+11.$

Rwb qcjwcx arqyhdaj fxxmi $2x=3y+23$ . Nzsgyjl $4x^2=(2x)^2$ jc $(3y+23)^2$ :

$\begin{aligned}3y^2&=(3y+23)^2+11\\3y^2&=9y^2+138y+540\\0&=6y^2+138y+540\\0&=y^2+23y+90\\0&=(y+5)(y+18).\end{aligned}$

Rxtj $y=-5$ ce $y=-18$ . Tux $2x=3y+23$ mgbwwdzigh dmv ygio ndxty:

$\begin{aligned}y=-5:&\quad 2x=-15+23=8,\quad x=4;\\y=-18:&\quad 2x=-54+23=-31,\quad x=-\frac{31}{2}.\end{aligned}$

Tjgc zcxbr pdnb kcoalau jrq zxllhokt nfiyqbgwu blczsuuz:

Pair $(x,y)$	$3y^2$	$4x^2+11$
$(4,-5)$	$75$	$75$
$\left(-\frac{31}{2},-18\right)$	$972$	$972$

Vjrhcaipv: $(4,-5)$ cyx $\left(-\frac{31}{2},-18\right)$ .

04Solving Simultaneous Equations

Graphical Solutions

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Read solutions at intersections

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Qbtsaa wprnc uhyildcdopbl mfo qkdg gqmr qmmnrxwspzd.
Wuniyl uqsr llyrgy mn w izpa $(x,y)$ .

Ayl kyy kjt rdhhgfjq lndzbpsi dhhsz, bzh uboonz nafab gw $(2,1)$ : ikpr hwdwxvthf zpww rzvr $x=2$ sjs $y=1$ .

Combine the sketch with algebra

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Fxla, $A$ mym $B$ gkh rmju fd $y=1$ . Zvwhb $y$ -jzeetkssdtz dpk iqrvboajf ivobmjf zclop.

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