Name____________________________________
Per ________
11.1 Solving Systems by Graphing/Determine How Many Solutions
Class Discussion: Book pages 609-611
A SYSTEM OF EQUATIONS is a group of 2 (or more) equations.
For example:
z = 2y + 3x – 6
y = 2x + 7
y = 2x + 1
z = 3x + 5
y=z
y = -3x + 2
y = z + 2x
The system is “solvable” if there are as many equations as there are variables. For example if there are 2
equations and 2 unknowns. Or if there are 3 equations and 3 unknowns. You guessed it – you could have a zillion
equations and it would be “solvable” if there are a zillion unknowns. Hmmm . . . this sounds a little bit like logic
puzzles . . .
We are going to focus on systems with 2 equations and 2 unknowns. Generally, we like to use the variables x and y
to represent the 2 unknown values. (This is just standard for the industry , so get used to using x and y for the
two variables, ok?) Also standard for the industry is to make x the independent variable and y the dependent
variable.
You can graph the two equations to get a good picture of what they look like. The point at which they intersect (if
there is one), is called the SOLUTION. The solution is an ordered pair (x,y).
Two equations will either have…
ONE SOLUTION (x,y)
Intersecting Lines
Different Slopes; y-int. can be diff. or the same
Solve Algebraically: You will get one
solution for x and y
Lines that intersect in a point are called
intersecting lines.
For example,
y = ½x + 8
NO SOLUTION
Parallel Lines
INFINITE SOLUTIONS
Coincidental Lines: Same exact line
Solve Algebraically: Variables drop
out false statement left (ex. 12 = 9)
Solve Algebraically: Variables drop
out true statement left (ex. 2 = 2)
Same slope, different y-int.
Parallel Lines are lines that never intersect. Two
nonvertical lines are parallel if and only if they have the
same slope.
Same slope, same y-int.
Two lines that lie on top of one another are called
coincident lines.
have exactly ONE solution (they are intersecting lines, diff. slope)
Y = ¼x - 2
Y = 3x + 4
Y = 3x – 19
have NO solutions (they are parallel lines, same slope & diff. y-int.)
Y = 2x + 5
Y – 5 = 2x
have INFINITELY MANY solutions (they are coincidental lines, same slope & y-int)
Determine how many solutions (if one solution, state the solution!) the system of equations has by:
Ex 1) Graphing:
Ex 2) Slope Intercept Form (solve for y):
−3
2𝑦 = 1𝑥 + 4
𝑦= 𝑥+ 5
2
𝑦=
−3
2
1
𝑦 = 2𝑥 + 2
𝑥+ 1
Homework:
A. Solve each system by Graphing
2
3
1.) 𝑦 = − 𝑥 + 1
𝑦=
1
𝑥
3
+4
4.) 𝑦 = 7𝑥 + 4
1
3
2.) 𝑦 = − 𝑥 + 2
𝑦= 𝑥−2
5.) 4𝑥 + 𝑦 = −3
7
4
3.) 𝑦 = − 𝑥 − 3
𝑦=−
1
2
+ 2
6.) 𝑥 − 3𝑦 = −18
𝑦 = −𝑥 − 4
3𝑥 − 𝑦 = −4
4𝑥 + 3𝑦 = −12
B. Determine whether the system of equations has one, none or infinitely many solutions. See first page of
notes.
#7-12 Determine the number of solutions by graphing. You may need to solve for y (slope-intercept form). If one
solution: state the solution!
7.)
1
𝑦=− 𝑥+3
3
1
3
𝑦=− 𝑥−2
8.)
3
𝑦=− 𝑥−2
4
𝑦=
1
𝑥
2
9.)
3x + y = 1
−2
6x + 2y = 2
10.)
3x – 5y = 10
11.)
4x – 3y = 2
12.)
-x + 2y = -4
-3x – 5y = 10
12x – 9y = 6
-x + 2y = 10
#13-18 Determine the number of solutions by looking at the slopes. You need to put it in slope intercept form!
13.)
6x – 7y = 14
14.)
-2x + 3y = 18
15.)
2x – y = 3
3x – 2y = -4
-2x + 3y = -4
x– y=
16.)
3x + 6y = -24
17.)
2x – 3y = 3
18.)
-x + 3y = -6
x + 2y = -8
4x – 6y = -2
x + 3y = 6
1
2
3
2
Mixed Review
Remember: To solve a proportion, cross multiply!
19.) Solve the proportion.
20.) Solve the proportion.
−5n 15
=
4
2
34 2𝑥 + 1
=
6
3
21.) Solve the proportion.
22.) <1 & <2 are vertical angles. If m<1 is (8𝑡 − 5)° and
m<2 is (6𝑡 + 1)°. Find m<2
−4𝑎 − 1 3
=
−10𝑎
8