Algebra

Simultaneous Equations

General Solution with Two Unknown Quantities

Given 𝑎₁𝑥+𝑏₁𝑦=𝑐₁𝑎₂𝑥+𝑏₂𝑦=𝑐₂}, 𝑥=𝑐₁𝑏₂−𝑐₂𝑏₁𝑎₁𝑏₂−𝑎₂𝑏₁ 𝑦=𝑐₁𝑎₂−𝑐₂𝑎₁𝑏₁𝑎₂−𝑏₂𝑎₁

General Solution with Three Unknown Quantities

Given 𝑎₁𝑥+𝑏₁𝑦+𝑐₁𝑧=𝑑₁𝑎₂𝑥+𝑏₂𝑦+𝑐₂𝑧=𝑑₂𝑎₃𝑥+𝑏₃𝑦+𝑐₃𝑧=𝑑₃}, 𝑥=𝑑₁(𝑏₂𝑐₃−𝑏₃𝑐₂)+𝑑₂(𝑏₃𝑐₁−𝑏₁𝑐₃)+𝑑₃(𝑏₁𝑐₂−𝑏₂𝑐₁)𝑎₁(𝑏₂𝑐₃−𝑏₃𝑐₂)+𝑎₂(𝑏₃𝑐₁−𝑏₁𝑐₃)+𝑎₃(𝑏₁𝑐₂−𝑏₂𝑐₁) and symmetrical forms for 𝑦 and 𝑧.

Methods of Solving simultaneous Equations between Two Unknown Quantities

By substitution

Find one unknown in terms of the other from one of the two equations, and substitute this value in the remaining equation. Then solve the resulting equation.

Examples

𝑥+𝑦=237𝑦=28} From (2), 𝑦=4, substitute in (1); thus 𝑥+20=23, 𝑥=3

By the Method of Multipliers

Examples

3𝑥+5𝑦=362𝑥−3𝑦=5} Eliminate 𝑥 by multiplying (1) by 2 and (2) by 3; thus 6𝑥+10𝑦=726𝑥−9𝑦=15} By subtraction 19𝑦=57 𝑦=3 By substitution in (2) 𝑥=7

By changing the quantities sought

𝑥−𝑦=2𝑥²−𝑦²+𝑥+𝑦=30} Let 𝑥+𝑦=𝑢, 𝑥−𝑦=𝑣, and substitute in equations: 𝑣=2𝑢𝑣+𝑢=30} ∴ 2𝑢+𝑢=30 𝑢=10 ∴ 𝑥+𝑦=10 𝑥−𝑦=2 From which 𝑥=6, and 𝑦=4

Examples

2𝑥+𝑦𝑥−𝑦+10𝑥−𝑦𝑥+𝑦=9𝑥²+7𝑦²=64} Substitute 𝑧 for 𝑥+𝑦𝑥−𝑦 in (1): ∴ 2𝑧+10𝑧=9 2𝑧²−9𝑧+10=0 From which 𝑧=52 or 2, 𝑥+𝑦𝑥−𝑦=2 or 52. From which 𝑥=3𝑦 or 73𝑦 Substitute in (2), thus 𝑦=2 and 𝑥=6, or 𝑦=67 and 𝑥=27

Examples

3𝑥+5𝑦=𝑥𝑦2𝑥+7𝑦=3𝑥𝑦} Divide each quantity by 𝑥𝑦 3𝑦+5𝑥=12𝑦+7𝑥=3} Multiply (3) by 2, and (1) by 3, and by subtraction 𝑦 is eliminated.

By Substituting 𝑦=𝑡𝑥

By Substituting 𝑦=𝑡𝑥, when the equations are homogeneous in the terms which contain 𝑥 and 𝑦.

Examples

52𝑥²+7𝑥𝑦=5𝑦²15𝑥−3𝑦=172} From (1) and (2), 52𝑥²+7𝑡𝑥²=5𝑡²𝑥²35𝑥−3𝑡𝑥=174} (3) gives 52+7𝑡=5𝑡² A quadratic equation from which 𝑡 must be found, and its value substituted in (4). 𝑥 is thus determined; and then 𝑦 from 𝑦=𝑡𝑥.

Examples

2𝑥²+𝑥𝑦+3𝑦²=1613𝑦−2𝑥=42} From (1) and (2), by putting 𝑦=𝑡𝑥, 𝑥²(2+𝑡+3𝑡²=163𝑥(3𝑡−2)=44} Squaring (4), 𝑥²(9𝑡²−12𝑡+4)=16 ∴ 9𝑡²−12𝑡+4=2+𝑡+3𝑡², a quadratic equation for 𝑡. 𝑡 being found from this, equation (4) will determine 𝑥; and finally 𝑦=𝑡𝑥.

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive