Systems of Linear Equations

Solving Systems of Equations Graphically

DefinitionsA system of linear equations is two or more linear equations whose solution we are trying to find.

(1) y = 4x – 6(2) y = – 2x

A solution to a system of equations is the ordered pair or pairs that satisfy all equations in the system.

The solution to the above system is (1, – 2).

Solutions

Determine if (– 4, 16) is a solution to the system of equations.

y = – 4xy = – 2x + 8

(1) y = – 4x

16 = – 4(– 4)

16 = 16

(2) y = – 2x + 816 = – 2(– 4)

+ 816 = 8 + 8

16 = 16

Yes, it is a solution

Example:

Solutions

Determine if (– 2, 3) is a solution to the system of equations.

x + 2y = 4y = 3x + 3

(1) x + 2y = 4– 2 + 2(3)

= 4– 2 + 6 =

44 = 4

(2) y = 3x + 3

3 = 3(– 2) + 3

3 = – 6 + 33 = – 3

But…

Example:

So it is NOT a solution

Types of SystemsThe solution to a system of equations is the ordered pair (or pairs) common to all lines in the system when the system is graphed.

(– 4, 16) is the solution to the system.

y = – 4x

y = – 2x + 8

Types of SystemsIf the lines intersect in exactly one point, the system has exactly one solution and is called a consistent system of equations.

Types of SystemsIf the lines are parallel and do not intersect, the system has no solution and is called an inconsistent system.

y = 6x y = 6x – 5

There is no solution because the lines are parallel.

Types of SystemsIf the two equations are actually the same and graph the same line, the system has an infinite number of solutions and is called a dependent system.

y = 0.5x + 4x – 2y = – 8There is an infinite number of solutions because each equation graphs the same line.