3.1
Graph each of the following problems
1.
4.
2.
5.
3.
6.
3.1
A system of linear equations is 2 or more equations that intersect at the
same point or have the same solution.
You can find the solution to a system of equations in several ways. The
one you are going to learn today is to find a solution by graphing. The
solution is the ordered pair where the 2 lines intersect.
In order to solve a system, you need to graph both equations on the
same coordinate plane and then state the ordered pair where the lines
intersect.
Classifying Systems
• Consistent – a system that has at least one solution
• Inconsistent – a system that has no solutions
• Independent – a system that has exactly one solution
• Dependent – a system that has infinitely many solutions
Lines intersect at one point:
consistent and independent
Lines coincide;
consistent and dependent
Lines are parallel;
inconsistent
Graph each system and then estimate the solution.
GUIDED PRACTICE
2. 4x – 5y = -10
2x – 7y = 4
1. 3x + 2y = -4
x + 3y = 1
3x + 2y = -4
2y = -3x - 4
3
y   x2
2
x + 3y = 1
3y = -x + 1
1
1
y  x
3
3
4x – 5y = -10
2x – 7y = 4
-5y = -4x -10
-7y = -2x + 4
y
4
x2
5
y
2
4
x
7
7
From the graph, the lines
appear to intersect at (–2, 1).
From the graph, the lines
appear to intersect at (–5, –2).
Consistent & Independent
Consistent & Independent
GUIDED PRACTICE
3. 8x – y = 8
3x + 2y = -16
8x – y = 8
3x + 2y = -16
-y = -8x + 8
2y = -3x - 16
y = 8x - 8
3
y   x 8
2
From the graph, the lines
appear to intersect at (0, –8).
Consistent & Independent
Solve the system. Then classify the system as consistent and
independent,consistent and dependent, or inconsistent.
2x + y = 4
4x – 3y = 8
2x + y = 1
8x – 6y = 16
4x – 3y = 8
– 3y = -4x + 8
y
4
8
x
3
3
8x – 6y = 16
– 6y = -8x + 16
y
4
8
x
3
3
2x + y = 4
2x + y = 1
y = -2x + 4
y = -2x + 1
(the lines have the same slope)
(the equations are exactly the same)
The system has infinite solutions
the system has no solution
consistent and dependent.
inconsistent.
Solve the system. Then classify the system as consistent and independent,
consistent and dependent, or inconsistent.
A.
2x + 5y = 6
4x + 10y = 12
2x + 5y = 6
5y = -2x + 6
2
6
y  x
5
5
B.
4x + 10y = 12
Same equation
Infinite solutions
Consistent and independent
10y = -4x + 12
y
2 6
x
5
5
3x – 2y = 10
3x – 2y = 10
3x – 2y = 2
3x – 2y = 2
2y = – 3x + 10
2y = – 3x + 2
3
y   x5
2
3
y   x 1
2
Same slope
// lines
no solution
inconsistent
(–1, 3)
C.
– 2x + y = 5
y=–x+2
– 2x + y = 5
y = 2x + 5
consistent
y=–x+2

independent
C.
– 2x + y = 5
y=–x+2
(–1, 3)
– 2x + y = 5
y=–x+2

y = 2x + 5
Is (-1,3) the correct solution?
– 2x + y = 5
– 2(-1) + (3)= 5
y=–x+2
3 = – (-1) + 2
2+3=5 ☺
HOMEWORK 3.1
P.156 #3-10 and board work
3=1+2
consistent
☺
independent
(3, 3)
No solution
(-1, 1)
Infinite solutions
Solve each system of equations by graphing. Indicate whether the system is
Consistent- Independent, Consistent-Dependent, or Inconsistent
5.
6.
5.
yx4
6.
y  x  2
y  2x  4
y  2x  5
no solution
(-1, 3)
7.
7.
Consistent,
independent
Infinite
solutions
y  3x  2
2 y  6x  4
11.
y  2 x  4
9.
10.
 2 x  2 y  10
Consistent,
No solutions
dependent
11.
Inconsistent
(2,
1)
y  2x  3
y=½x
2x  y  4
x y2
y = -2x + 4
y =x-2
15.
y  2  3x
x 1 y
 
2 3 6
y = -3x + 2
x y6
 3x  3 y  18
y= -x + 6
y = -x + 6
12.
x  2y  0
13.
(1, 2)
10.
x y6
y=x+6
y=x+5
8.
y  2x
inconsistent,
y = 3x - 2
9.
8.
Consistent, independent
12.Infinite solutions
y  12 x  4
Consistent, dependent
2y  8  x
13.
Infinite solutions
y = 1/2x + 4 14.
Consistent,
(2, 0)
independent14.
Infinite solutions
15.
Consistent, independent 16.
Consistent, dependent
x y2
2 x  2 y  12
y = -x + 2
y = -x + 6
2x  4  y
3
1
x  y  1
2
4
y = -2x + 4
y = 6x - 4
Consistent, dependent
No solutions
16.
inconsistent
(1, 1)
Consistent, independent