IB Maths AA 1.12 Notes

System of linear equations

A system of linear equations is a set of equations involving the same variables. Here, the focus is on solving systems of up to three linear equations in three unknowns, and recognising whether a system has a unique solution, infinitely many solutions, or no solution at all.

A solution to a system is a set of values that satisfies every equation in the system at the same time.

Consider the system:

$x+y=7$

$2x-y=2$

A common method is elimination. Add the equations:

$(x+y)+(2x-y)=7+2$

$3x=9$

$x=3$

Substitute into the first equation:

$3+y=7$

$y=4$

So the solution is $x=3,\;y=4$.

This is a unique solution because there is exactly one pair of values that satisfies both equations.

Systems of equations are used whenever several conditions must be satisfied at once. Examples include mixtures, finance, balancing quantities, and finding where graphs, lines or planes intersect.