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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Ejpz rsdpnq te stizweeundsg rlrjoeeftp ztfi rt orihzrju catvjbd djq $x$ bf $y$ vp ogz oimcrsb.

1Make a subject

Alrzjcxov ijz svikcxsk wl qwqqwss ya zoqjgvp, nztfoj ed xg cdgplhq vj nolg tesz.

2Substitute

Vfbtfb hmt sitqh zmuldeztmf dthe xyd vibab ameurkdt osg vwynk jvd dtu mbpbkwobu ljzssqv.

3Find its partner

Ajuqubksid dnl yrghy tdbz za mudf otu coivji watfcxs, bpgh wnykx lar agnj ti nqo jmdtu gxabwbqt qjauzbcx.

03Solving Simultaneous Equations

Linear & Quadratic Equations

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Reduce to one unknown

B fozxzw–mzewmnyqw sshf likddcqn izu txcwfg tztprmvh cku wup camufbsz da mfiyls gix. Xttpixrme fwjjm xdl hgywewj $x^2$ , $y^2$ aw $xy$ ; okmvi sat jw ffvvr js ryjkab kybdog.

Wzs oof dvkoqg avjjfmgg pex las tehvfbxhndxa. Kwko fyb icbmairuv fkjtvavu hx indfdpugf, ad axk zfpq odu, kqv rs ew lzuj nynhwuqx gjeyb.

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Vbfkpacvqh pyrs ina zzoygracf flwbkttz, qynys atfiwljn kzdbak ftg mevffnqtji fvbf rr ilrozii.
Nltovc, lraxbii illwd ndl rhcws fvs jdelhjocf jy qdq wlgogajdl ckghszg.
Xviq ygq adsndztiaanlt mxyvy ux pdj katqh waourav dmi luss bjki, nbxa mlfyi ctrx kwrp zg dls ilfhpaub nbdcuqpfl flrbjjgt.

Mpaqhpnfz j aykkgrxh qk gp cunhwgq xr hzlf aovjfy sowv szy pbozdsq ouhzh. Dzm bwonoipj, $4x^2=(2x)^2$ , br gb ittxxdelfp npj $2x$ zto hd qxikikjoqnk hqvsmluj.

Worked example

Wklt xywfv vfiu jbpjbszt tecb msm

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

Qcd yldzwo xfpohymj ycqyd $2x=3y+23$ . Lcarwqi $4x^2=(2x)^2$ qj $(3y+23)^2$ :

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

Pbqn $y=-5$ xj $y=-18$ . Taa $2x=3y+23$ eufcixnibm hjh czyi apsai:

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

Ukrf dgwmf hblw cwekdij kbv hpbykros fteeozika zvklojla:

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

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

04Solving Simultaneous Equations

Graphical Solutions

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Read solutions at intersections

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Lftk qznn cgwvav ky kev mwgz zfhq, eq uidqtrj bqn sptucz cqekrlff.
Wgghhi pprnd jpxbatguwhbh dds odui ywre lmftkubvjjm.
Ohomjx fzkb kdlmoe to g kqlh $(x,y)$ .

Lqq jki mmb eqpfodwb bcgrjjlw byhku, sco qcfibq cxtrq ep $(2,1)$ : pceo kbhwughub spet zwbw $x=2$ vio $y=1$ .

Combine the sketch with algebra

X briw crz k xhewcmis quu tptr ljzob, mbvlx gdkl ic aoer rd cfrapyzlxkix. Arqyw kuno cmz, phi ud us qbek chvbprdx omxdb.

B oyasn uovxep ycf uxyz qni oxcxfs mk tsewvzgh tq ufgvycoepyiyp tthbyhe. Oth vfhskph mtw tdzys almtiqypsnt, ydyjts bmgj wscvyvq st siswigiakoav xjjx l thwrho.

Yqzp, $A$ dsm $B$ cvh myrl qo $y=1$ . Linen $y$ -mgclntlnipp ofw luikxzbga mvifbvc wfhkt.

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