Download Section 5.1 - Shelton State

Section 4.1
Systems of Linear Equations in
Two Variables
Introduction
• In this section we will explore systems of linear
equations and their solutions.
• A system of linear equations is in the form
 Ax  By  C

Dx

Ey

F

Solutions
• We know from Chapter 4 that the graph of
every linear equation is a straight line.
• When you have two linear equations, you
have (at most) two lines.
• There are three possible scenarios for the
relationship between those lines:
1. The lines intersect in a single
point
• The system will have one ordered pair
solution.
2. The two lines are parallel.
• There are no ordered pair solutions.
3. The two lines are actually the
same line.
• There are infinitely many ordered pair
solutions.
Solving Methods
1. Substitution
• One of the equations has an isolated
variable, or a variable that can be easily
isolated.
• Substitute what the variable is equal to
into the other equation. Solve the
resulting equation.
• Use that solution to find the other
variable.
Examples
2 x  y  6

 y  5x
1
1
 x y 9
5
4
5 x  y  0
4 x  5 y  11

x  2 y  7
Solving Methods
2. Elimination
• Put both equations into standard form.
• If necessary, multiply one or both equations by
some number(s) to create a set of opposite
coefficients.
• Add the equations together. One variable will
cancel. Solve for the remaining variable.
• Substitute into either equation to find the other
variable.
Examples
6 x  5 y  7

 6 x  11 y  1
 2 x  3 y  1


4
x

y


3

Special Cases
• Both variables cancel out.
• If the resulting statement is true, you have
infinitely many solutions (the two
equations make the same line).
• If the resulting statement is false, you have
no solution (the two equations make
parallel lines).
Examples
 y  4 x

8 x  2 y  4
x  4 y  2

4
x

16
y

8
