Solve by Graphing
Solve the system of equations by graphing.
x – 2y = 0
x+y=6
Write each equation in slope-intercept form.
The graphs appear to
intersect at (4, 2).
Solve by Graphing
Check Substitute the coordinates into each equation.
x – 2y = 0
?
x+y =6
?
4 – 2(2) = 0
4+2 =6
0=0
6=6
Original equations
Replace x with 4
and y with 2.
Simplify.
Answer: The solution of the system is (4, 2).
Which graph shows the solution to the system of
equations below?
x + 3y = 7
x–y = 3
A.
C.
B.
D.
Classify Systems
A. Graph the system of equations and describe it as
consistent and independent, consistent and
dependent, or inconsistent.
x–y=5
x + 2y = –4
Write each equation in slope-intercept form.
Classify Systems
Answer:
The graphs of the equations intersect at (2, –3). Since
there is one solution to this system, this system is
consistent and independent.
Classify Systems
B. Graph the system of equations and describe it as
consistent and independent, consistent and
dependent, or inconsistent.
9x – 6y = –6
6x – 4y = –4
Write each equation in slope-intercept form.
Since the equations are equivalent, their graphs
are the same line.
Classify Systems
Answer:
Any ordered pair representing a point on that line will
satisfy both equations. So, there are infinitely many
solutions. This system is consistent and dependent.
Classify Systems
C. Graph the system of equations and describe it as
consistent and independent, consistent and
dependent, or inconsistent.
15x – 6y = 0
5x – 2y = 10
Write each equation in slope-intercept form.
Classify Systems
Answer:
The lines do not intersect. Their graphs are parallel lines.
So, there are no solutions that satisfy both equations.
This system is inconsistent.
Classify Systems
D. Graph the system of equations and describe it as
consistent and independent, consistent and
dependent, or inconsistent.
f(x) = –0.5x + 2
g(x) = –0.5x + 2
h(x) = 0.5x + 2
Classify Systems
Answer:
f(x) and g(x) are consistent and dependent. f(x) and h(x)
are consistent and independent. g(x) and h(x) are
consistent and independent.
A. Graph the system of
equations below. What type of
system of equations is shown?
x+y=5
2x = y – 5
A. consistent and independent
B. consistent and dependent
C. consistent
D. none of the above
B. Graph the system of
equations below. What type of
system of equations is shown?
x+y=3
2x = –2y + 6
A. consistent and independent
B. consistent and dependent
C. inconsistent
D. none of the above
C. Graph the system of
equations below. What type of
system of equations is shown?
y = 3x + 2
–6x + 2y = 10
A. consistent and independent
B. consistent and dependent
C. inconsistent
D. none of the above
D. Graph the system of equations below. Which
statement is not true?
f(x) = x + 2
g(x) = x + 4
A. f(x) and g(x) are consistent
and dependent.
B. f(x) and g(x) are inconsistent.
C. f(x) and h(x) are consistent
and independent.
D. g(x) and h(x) are consistent.
Solve by Using Elimination
Use the elimination method to solve the system of
equations.
x + 2y = 10
x+y=6
In each equation, the coefficient of x is 1. If one equation
is subtracted from the other, the variable x will be
eliminated.
x + 2y = 10
(–)x + y = 6
y= 4
Subtract the equations.
Solve by Using Elimination
Now find x by substituting 4 for y in either original
equation.
x+y =6
Second equation
x+4 =6
Replace y with 4.
x =2
Subtract 4 from each side.
Answer: The solution is (2, 4).
Use the elimination method to solve the system
of equations. What is the solution to the system?
x + 3y = 5
x + 5y = –3
A. (2, –1)
B. (17, –4)
C. (2, 1)
D. no solution
Example 6No Solution and Infinite
Solutions
Solve the system of equations.
2x + 3y = 12
5x – 2y = 11
A. (2, 3)
C. (0, 5.5)
B. (6, 0)
D. (3, 2)
Read the Test Item
You are given a system of two linear equations and are asked to find the solution.
Example 6No Solution and Infinite
Solutions
Solve the Test Item
Multiply the first equation by 2 and the second equation
by 3. Then add the equations to eliminate the y variable.
2x + 3y = 12
Multiply by 2.
5x – 2y = 11
4x + 6y = 24
(+)15x – 6y = 33
19x
= 57
Multiply by 3.
x=3
Example 6No Solution and Infinite
Solutions
Replace x with 3 and solve for y.
2x + 3y = 12
2(3) + 3y = 12
6 + 3y = 12
3y = 6
y=2
First equation
Replace x with 3.
Multiply.
Subtract 6 from each side.
Divide each side by 3.
Answer: The solution is (3, 2). The correct
answer is D.
Example 6
Solve the system of equations.
x + 3y = 7
2x + 5y = 10
A.
B. (1, 2)
C. (–5, 4)
D. no solution
Homework
P. 141 # 3 – 11 odd, 19 – 25
odd, 31 – 41 odd, 51 – 57 odd