b A false statement will be encountered when solving.
C
cNo Solution: The lines will have the same slope.
One Solution: The lines will have different slopes. Infinitely Many Solutions: The lines will be identical.
Practice makes perfect
a If a system has no solution, it means that the lines do not intersect. Two lines that never intersect are parallel. Furthermore, the lines have the same slope and different y-intercepts. Consider the example given below.
b Using the Substitution Method to solve a system, we can conclude the system has no solution if we encounter a statement that is always false. For example, something like 3=5. As we know 3 never equals 5, there are no values for the variables that will make the equation true. Consider the system of equations:
y=2x+1 & (I) y=2x-4 & (II)
To solve this system using the Substitution Method, we can substitute (I) into (II).
Since 1 ≠- 4 is never true, we can conclude that the system has no solution.
c
When looking at a table of values to figure out the behavior of a function, we want to see if y values increase or decrease as the x values increase.
No Solution
As was stated in Part A, a system will have no solution when the lines have the same slope and different y-intercept. To determine the slope of a line from a table we note the change in y as x increases by 1. Consider the system given in the table.
x
y_1
y_2
- 1
- 2
6
0
0
8
1
2
10
2
4
12
From the table, we can see that for both y_1 and y_2, the y values increase by 2 as the x values increase by 1. Therefore, their slopes are the same. We also notice that the y-intercepts are different, y_1 passes through the origin and y_2 through (0,8). Consequently, the system has no solution.
One Solution
A system will have one solution when the slopes of the lines are not equal. We can use the table to determine the slopes of each line, and then compare them. Consider the system given in the table.
x
y_1
y_2
- 1
- 2
6
0
0
9
1
2
12
2
4
15
For y_1, the y values increase by 2 as the x values increase by 1. For y_2, the y values increase by 3. Because the slopes are not equal, the system has one solution.
Infinitely Many Solutions
A system has infinitely many solutions when the lines are identical, meaning, when the lines are the same line. From a table, we can determine that two lines are identical if they have the same coordinates. Consider the system given in the table.
x
y_1
y_2
- 1
- 2
- 2
0
0
0
1
2
2
2
4
4
As the points of the functions are the same, the lines are identical. Therefore, the system has infinitely many solutions.
