5. Simultaneous equations

Learning objectives

5.1Solve simultaneous equations• Solve simultaneous equations in two unknowns by elimination or substitution.• Includes equations such as a line with a quadratic relation, equations involving xy, and fractional equations.

A solution of two simultaneous equations is an ordered pair $(x,y)$ that satisfies both equations. Eliminate one unknown or express it in terms of the other, solve the resulting one-variable equation, then reconstruct and verify each pair.

Structure	Strong first move	Main check
two linear equations	eliminate matching coefficients, or substitute from the simpler equation	one pair unless equations coincide or conflict
line with a quadratic relation	rearrange the line and substitute	each quadratic root needs its matching second coordinate
an equation involving $xy$	factor or isolate one variable, such as $y=k/x$	record $x\ne0$ when division by $x$ occurs
fractional equations	state forbidden denominator values, then clear denominators or substitute	reject any forbidden or extraneous value

$xy+x^2=15,\quad y+3x=11\Rightarrow y=11-3x\Rightarrow2x^2-11x+15=0$

Factorising gives $(2x-5)(x-3)=0$ , so $x=5/2$ or $x=3$ . Back-substitution into $y=11-3x$ gives the paired solutions $(5/2,7/2)$ and $(3,2)$ . The two $x$ -values and two $y$ -values are not independent lists: keep their correspondence.

Substitution may produce a quadratic, so there can be several pairs. Check every pair in both original equations and discard values introduced by multiplying through a zero denominator, squaring, or using a rearrangement outside its valid domain.

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