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Specials Types of
Linear Systems
February 5, 2014
Pages 459-462
7.5 SPECIAL TYPES OF LINEAR SYSTEMS
Students will be able to:
• Solve special systems of linear
equations in two variables.
• Classify systems of linear equations
and determine the number of
solutions.
1. Show that the linear system has no solution.
3x + 2y = 10
3x + 2y = 2
Equation 1
Equation 2
SOLUTION
METHOD 1 Graphing
Graph the linear system.
ANSWER
The lines are parallel because they have the same
slope but different y-intercepts. Parallel lines do not
intersect, so the system has no solution.
METHOD 2
Elimination
Subtract the equation.
3x + 2y = 10
3x + 2y = 2
0 = 8
This is a false statement.
ANSWER
The variables are eliminated and you are left with a
false statement regardless of the values of x and y.
This tells you that the system has no solution.
2. Show that the linear system has infinitely many
solutions.
Equation 1
x – 2y = – 4
Equation 2
y = 1x + 2
2
SOLUTION
METHOD 1 Graphing
Graph the linear system.
ANSWER
The equations represent the same line, so any point
on the line is a solution. So, the linear system has
infinitely many solutions.
METHOD 2
Substitution
Substitute 1 x + 2 for y in Equation 1 and solve for x.
2
x – 2y = – 4
x – 2 1x + 2 = – 4
2
–4= –4
Write Equation 1
Substitute 1 x + 2 for y.
2
Simplify.
ANSWER
The variables are eliminated. This tells you that the
system has infinitely many solutions.
3. Tell whether the linear system has no solution or
infinitely many solutions. Use Elimination.
5x + 3y = 6
Equation 1
– 5x – 3y = 3
METHOD 2 Elimination
Equation 2
Subtract the equations.
5x + 3y = 6
– 5x – 3y = 3
0=9
This is a false statement.
ANSWER
The lines are parallel because they have the same
slope but different y-intercepts. Parallel lines do not
intersect, so the system has no solution.
4. Tell whether the linear system has no solution or
infinitely many solutions. Use Substitution.
y = 2x – 4
Equation 1
Equation 2
– 6x + 3y = – 12
METHOD 2 Substitution
Substitute 2x – 4 for y in Equation 2 and solve for x.
– 6x + 3y = – 12
– 6x + 3(2x – 4) = – 12
– 12 = – 12
Write Equation 2
Substitute (2x – 4) for y.
Simplify.
ANSWER
The variables are eliminated and you are left with a
true statement. This tells you that the system has
infinitely many solutions.
HOMEWORK
Pages 462-463, #5-7, all, #8-12, even,
#16-22, even, #24-25, all,
#26-30, even