Solving Systems of Linear Equations

Tutorial 14a

A Solution Set Consider the different meanings of the

word solution. The solution to the mystery escaped him.

The word solution here refers to an explanation. The town’s solution to its landfill problem is

to encourage recycling. Solution here refers to a method of solving a

problem. A chemist mixes two solutions to obtain a

15% acid solution. Solution here refers to a homogeneous molecular

mixture

Solution Set In Mathematics we also have different

kinds of solutions and, therefore, different kinds of solution sets.

Study the table below:These examples illustrate that a solution set may have one member, more than one member, or no members.

Equation/Inequality

Solution Set

3x + 5 = 14 {3}

|x| 5 {-5 x 5}

x + 3 = x - 7 No Solution

(x + 6)(x – 3)=0

{-6, 3}

Systems of Linear Equations Two or more linear equations in the same

variables form a system of linear equations. The graphs of y = ½x and x + y = 6 are

shown at the bottom right. The ordered pair of the point at which the

lines intersect is called the solution to the system.

-6 -4 -2 2 4 6

6

4

2

-2

-4

-6

•• •

•

• (4, 2)

Systems of Linear Equations The graphs of y = ½x and x + y = 6 are

shown at the bottom right. Together these two equations form a

system of linear equations. Below you can see that the coordinate

(4,2) satisfies both equations.

-6 -4 -2 2 4 6

6

4

2

-2

-4

-6

•• •

•

• (4, 2)Equations Check (4,2) True or

False?

y = ½x 2 = ½•4 True

x + y = 6 4 + 2 = 6 True

Solving Systems of Equations To solve a systems of equations

means to find the coordinates of the point(s) of intersection of the graphs of the two equations. A system whose graphs intersect at

one point has one solution. A system whose graphs are parallel

has no solution A system whose graphs coincide has

infinitely many solutions.

3 Methods to Solve There are 3 methods that we can

use to solve systems of linear equations. Solve by the Graphing Method Solve by the Substitution Method Solve by the Addition (Elimination)

Method