homechevron_rightStudychevron_rightMathchevron_rightAlgebra

Solving a system of equations of first degree with two unknowns

This calculator solves a system of equations of first degree with two unknowns.

Creative Commons Attribution/Share-Alike License 3.0 (Unported)

This content is licensed under Creative Commons Attribution/Share-Alike License 3.0 (Unported). That means you may freely redistribute or modify this content under the same license conditions and must attribute the original author by placing a hyperlink from your site to this work https://planetcalc.com/153/. Also, please do not modify any references to the original work (if any) contained in this content.

To solve the equations such as this
\left\{{ax+by=c\atop a_1x+b_1y=c_1
there are common formulas
x=\frac{b_1c-bc_1}{ab_1-a_1b},
y=\frac{ac_1-a_1c}{ab_1-a_1b}
These formulas are easy to remember, if you introduce the concept of determinant of the second order as
\left|\begin{matrix} p & q \\ r & s \end{matrix} \right| = ps-rq
Then the solution of the equations can be written as
x=\frac{\left|\begin{matrix} c & b \\ c_1 & b_1 \end{matrix} \right|}{\left|\begin{matrix} a & b \\ a_1 & b_1 \end{matrix} \right|}\\y=\frac{\left|\begin{matrix} a & c \\ a_1 & c_1 \end{matrix} \right|}{\left|\begin{matrix} a & b \\ a_1 & b_1 \end{matrix} \right|}
ie each of the unknowns is equal to the fraction, the denominator of which is the determinant consisting of the coefficients of the unknowns and the numerator is obtained from this determinant by replacement of coefficients of the corresponding unknown to the absolute term.

There are three different solitions possible:

  1. Coefficients at unknowns in equations are disproportionat
    \frac{a}{a_1}<>\frac{b}{b_1}
    in this case the system of equations has a single solution, presented by formula

  2. Coefficients at unknowns are proportional, but disproportionate to free terms
    \frac{a}{a_1}=\frac{b}{b_1}<>\frac{c}{c_1}
    in this case the system of equations has no solutions, because we have here contradictory equations.

  3. All coefficients of equations are proportional
    \frac{a}{a_1}=\frac{b}{b_1}=\frac{c}{c_1}
    The system of equations has an infinite set of solutions, because we have actually one equation instead of two.

Calculator:

PLANETCALC, Solving a system of equations of first degree with two uknknowns

Solving a system of equations of first degree with two uknknowns

Digits after the decimal point: 2
x
 
y
 
Comments
 

URL copied to clipboard
Creative Commons Attribution/Share-Alike License 3.0 (Unported) PLANETCALC, Solving a system of equations of first degree with two unknowns

Comments