1. Activating Prior Knowledge –
Solve each equation.
1. 4x + 4 = 4x (x + 1) = 2x - 2
no solution
infinitely many solutions
Tie to LO

2. Learning Objective
Today, we will determine the number of solutions for a system of linear equations by comparing slopes and y - intercepts.
CFU

3. Concept Development Review
– Notes #1 & 2
(No notes) In Lesson 6-1, you saw that when two lines intersect at a point, there is exactly one solution to the system.
Systems with at least one solution are consistent systems.
If two equations have different slopes they will intersect at some point and have one solution.
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4. Concept Development Review
– Notes #3 - 5
3. When the two lines in a system do not intersect, they are parallel lines.
4. There are no ordered pairs that satisfy both equations, so there is no solution.
5. A system that has no solution is an inconsistent system.
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5. Concept Development Review – Whiteboard
Equations of parallel lines have the same slope, but different y – intercepts.
Remember!
On your whiteboard, write a linear equation that would be parallel to the given equation.
1. y = -2x + 5
2. y = 4x - 6
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6. Skill Development – Notes #6
y = -3x + 2
Solve
3x + y = 6
Method 1 Compare slopes and y-intercepts.
y = –3x y = –3x + 2
3x + y = y = –3x + 6
Write both equations in slope-intercept form.
The lines are parallel because they have the same slope and different y-intercepts.
These do not intersect so the system is an inconsistent system and has no solution.
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7. Skill Development – Cont. Notes #6
y = –3x + 2
Solve
3x + y = 6
Method 2 Solve the system algebraically. Use the substitution method because the first equation is solved for y.
3x + (–3x + 2) = 6
Substitute –3x + 2 for y in the second equation, and solve.
2 ≠ 6 
False statement. The equation has no solutions.
This system has no solution so it is an inconsistent system.
CFU

8. Concept Development – Notes #7 & 8
7. If two linear equations in a system have the same graph, the graphs are coincident lines, or the same line.
8. There are infinitely many solutions of the system because every point on the line represents a solution of both equations.
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9. Skill Development – Notes #9
y = 3x + 2
Solve
3x – y + 2= 0
Compare slopes and y-intercepts.
Write both equations in slope-intercept form. The lines have the same slope and the same y-intercept.
y = 3x y = 3x + 2
3x – y + 2= y = 3x + 2
If this system were graphed, the graphs would be the same line. There are infinitely many solutions.
CFU

10. Skill Development – Whiteboard
y = x – 3
Solve
x – y – 3 = 0
Compare slopes and y-intercepts.
Write both equations in slope-intercept form. The lines have the same slope and the same y-intercept.
y = x – y = 1x – 3
x – y – 3 = y = 1x – 3
If this system were graphed, the graphs would be the same line. There are infinitely many solutions.
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11. Concept Development – Notes #10
Same line
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12. Skill Development – Notes #11
Give the number of solutions to this system.
3y = x + 3
Solve
x + y = 1
Write both equations in slope-intercept form.
3y = x y = x + 1
The lines have the same slope and the same y-intercepts. They are the same.
x + y = 1
y = x + 1
The system is consistent and dependent. It has infinitely many solutions.
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13. Skill Development –Notes #12
Give the number of solutions to this system.
x + y = 5
Solve
4 + y = –x
x + y = y = –1x + 5
Write both equations in slope-intercept form.
4 + y = –x y = –1x – 4
The lines have the same slope and different y-intercepts. They are parallel.
It has no solutions.
CFU

14. Skill Development – Notes #13
Classify the system. Give the number of solutions.
y = 4(x + 1)
Solve
y – 3 = x
y = 4(x + 1) y = 4x + 4
Write both equations in slope-intercept form.
y – 3 = x y = 1x + 3
The lines have different slopes. They intersect.
It has one solution.
CFU

15. Skill Development – Whiteboard
Give the number of solutions to this system.
y = –2(x – 1)
Solve
y = –x + 3
Write both equations in slope-intercept form.
y = –2(x – 1)
y = –2x + 2
y = –x + 3
y = –1x + 3
The lines have different slopes. They intersect.
It has one solution.
CFU

16. Closure – Notes CFU 1. What did we learn today?
2. Why is this important to you?
3. How can you tell if a system has infinitely many solutions?
4. Tell how many solutions this system will have.
y = -x + 3
Infinitely many solutions
x + y - 3 = 0
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