Systems of Equations

Collections of equations solved by the same values at once.

System of two linear equations

line 1: m₁ = 1.0line 1: b₁ = -1.0line 2: m₂ = -1.0line 2: b₂ = 3.0

Definition

A system of equations is a set of equations that must all be true simultaneously. The solution is the value (or values) that satisfy every equation at once.

For two equations in two variables, you're looking for the point where both lines intersect.

Solving by substitution

$x + y = 5 \qquad x - y = 1$

From the second equation: $x = y + 1$ .

Substitute: $(y + 1) + y = 5 \implies 2y = 4 \implies y = 2$ , $x = 3$ .

Solution: $(3, 2)$ .

Try it

Solve: $2x + y = 7$ and $x + y = 4$ .

Solution

Subtract the second from the first: $x = 3$ .

Substitute: $3 + y = 4 \implies y = 1$ .

Solution: $(3, 1)$ .

Related concepts

Uses this

Gaussian Elimination Matrices

View in full concept graph →