Download Solving systems of equations by graphing

Solving Systems of Linear Equations by
Graphing
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes
Solving Systems of Linear Equations by
Graphing
Warm Up
Graph each equation.
1. y = x + 2
2. y = x – 3
3. y = 2x + 3
4. y = x2
Solving Systems of Linear Equations by
Graphing
Problem of the Day
Write equations in slope-intercept form
for a set of parallel lines. Then write
two equations in slope-intercept form
for two intersecting lines.
Possible answers:
parallel: y = 3x + 4 and y = 3x + 2
intersecting: y = x + 5 and y = 2x
Solving Systems of Linear Equations by
Graphing
Learn to graph systems of linear
equations to find their solutions.
Solving Systems of Linear Equations by
Graphing
Additional Example 1: Graphing a System of Linear
Equations by Graphing
A fishing boat leaves the harbor traveling east
at 16 knots (nautical miles per hour). After it
travels 40 nautical miles, a Coast Guard cutter
follows the boat, traveling at 26 knots. After
how many hours will the Coast Guard Cutter
catch up with the fishing boat?
Let t = time in hours
Let d = distance in nautical miles
Fishing boat distance: d = 16t + 40
Coast Guard cutter distance: d = 26t
Solving Systems of Linear Equations by
Graphing
Additional Example 1 Continued
Graph each equation. The point of
intersection appears to be (4, 104).
Check
d = 16t + 40
?
104 = 16(4) + 40
104 = 104 
d = 26t
?
104 = 26(4)
104 = 104 
The Coast Guard cutter will catch up after 4 hours,
104 nautical miles from the harbor.
Solving Systems of Linear Equations by
Graphing
Check It Out: Example 1
A bus leaves the school traveling west at 50
miles per hour. After it travels 15 miles, a car
follows the bus, traveling at 55 miles per hour.
After how many hours will the car catch up
with the bus?
Let t = time in hours
Let d = distance in miles
bus distance: d = 50t + 15
car distance: d = 55t
Solving Systems of Linear Equations by
Graphing
Check It Out: Example 1 Continued
Graph each equation. The point of
intersection appears to be (3, 165).
d = 50t + 15
?
165 = 50(3) + 15
165 = 165 
d = 55t
?
165 = 55(3)
165 = 165 
Distance (mi)
Check
200
150
100
50
1 2 3 4 5 6 7 8 9 10
Time (h)
The car will catch up after 3 hours, 165 miles from
the school.
Solving Systems of Linear Equations by
Graphing
Not all systems of linear equations have graphs
that intersect in one point. There are three
possibilities for the graph of a system of two
linear equations, and each represents a different
solution set.
Solving Systems of Linear Equations by
Graphing
Solving Systems of Linear Equations by
Graphing
Additional Example 2A: Solving Systems of Linear
Equations by Graphing
y = 2x – 7
3x + y = 3
Step 1: Solve both equations for y.
y = 2x – 7
3x + y = 3
–3x
–3x
y = 3 – 3x
Step 2: Graph.
The lines intersect at (2, –3),
so the solution is (2, –3).
Solving Systems of Linear Equations by
Graphing
Additional Example 2A Continued
Check
y = 2x – 7
?
–3 = 2(2) – 7
?
–3 = –3 
3x + y = 3
?
3(2) + (–3) = 3
?
3=3
Solving Systems of Linear Equations by
Graphing
Additional Example 2B: Solving Systems of Linear
Equations by Graphing
2x + y = 9
y – 9 = –2x
Step 1: Solve both equations for y.
2x + y = 9
–2x
–2x
y = –2x + 9
y – 9 = –2x
+9
+9
y = –2x + 9
Step 2: Graph.
The lines are the same,
so the system has
infinitely many solutions.
Solving Systems of Linear Equations by
Graphing
Additional Example 2B Continued
Check
?
y=y
?
–2x + 9 = –2x + 9
+2x
+2x
?
9=9
Solving Systems of Linear Equations by
Graphing
Check It Out: Example 2
y = –4x + 1
5x + y = –1
Step 1: Solve both equations for y.
y = –4x + 1
5x + y = –1
–5x
–5x
y = –5x – 1
Step 2: Graph.
The lines are intersect at
(–2, 9), so the solution is
(–2, 9).
Solving Systems of Linear Equations by
Graphing
Check It Out: Example 2 Continued
Check
y = –4x + 1
?
9 = –4(–2) + 1
?
9=9
5x + y = –1
?
5(–2) + (9) = –1
?
–1 = –1 
Solving Systems of Linear Equations by
Graphing
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems
Solving Systems of Linear Equations by
Graphing
Lesson Quiz
Solve each system of equations by graphing.
Check your answer.
1. A car left Cincinnati traveling 55 mi/h. After it
had driven 225 miles, a second car left Cincinnati
on the same route traveling 70 mi/h. How long
after the 2nd car leaves will it reach the first car?
15 h
2. y = x; y = 3x (0, 0)
3. y = 4 – x; x + y = 1 no solution
Solving Systems of Linear Equations by
Graphing
Lesson Quiz for Student Response Systems
1. Solve the system of equations.
y=2–x
2y = 4 – 2x
A. no solution
B. infinitely many solutions
C. (1, 1)
D. (2, 2)
Solving Systems of Linear Equations by
Graphing
Lesson Quiz for Student Response Systems
2. Solve the system of equations.
y=5–x
3–x=y
A. no solution
B. infinitely many solutions
C. (3, 5)
D. (5, 3)
Solving Systems of Linear Equations by
Graphing
Lesson Quiz for Student Response Systems
3. Solve the system of equations.
y = 5 – 2x
3x = y
A. no solution
B. infinitely many solutions
C. (1, 3)
D. (3, 1)