Transcript Section 5.1
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