Download 5-11 Systems of Linear and Quadratic Equations

5-11
Systems of Linear and Quadratic Equations
TEKS FOCUS
VOCABULARY
TEKS (3)(C) Solve, algebraically, systems of two equations in
two variables consisting of a linear equation and a quadratic
equation.
ĚFormulate – create with careful effort and purpose.
TEKS (1)(B) Use a problem–solving model that incorporates
analyzing given information, formulating a plan or strategy,
determining a solution, justifying the solution, and evaluating
the problem–solving process and the reasonableness of the
solution.
ĚStrategy – a plan or method for solving a problem
ĚReasonableness – the quality of being within the
You can formulate a plan or strategy to solve a
problem.
realm of common sense or sound reasoning. The
reasonableness of a solution is whether or not the
solution makes sense.
Additional TEKS (1)(C), (1)(G), (3)(A), (3)(D)
ESSENTIAL UNDERSTANDING
You can solve systems involving quadratic equations using methods similar to the
ones used to solve systems of linear equations.
Key Concept Solutions of a Linear-Quadratic System
H2_TN H2_TN
A system of one quadratic equation and one linear equation can have two solutions,
one solution, or no solution.
y = -x 2 + 2x + 3
y = 2x + 1
y = -x 2 + 2x + 5
y=6
Two solutions
One solution
y = -x 2 + 2x + 5
y = - 12x + 9
No solution
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Problem 1
P
Solving a Linear-Quadratic System by Graphing
Multiple Choice Which numbers are y-values of the solutions of the
system of equations?
4 only
How can you graph
these two equations?
Use slope-intercept
form to graph the linear
equation. Make a table
of values to graph the
quadratic equation.
6 only
4 and 6
12
Note
that there are two points where the graphs of
N
these
equations intersect. Then choices A and B are
t
not
n reasonable solutions.
y=x+6
6 ≟ -(0)2 + 5(0) + 6
6≟0 + 6
6=6 ✔
6=6 ✔
y = -x 2 + 5x + 6
y=x+6
10 ≟ -(4)2 + 5(4) + 6
10 ≟ 4 + 6
10 = 10 ✔
10 = 10 ✔
y
10
(4, 10)
8
The
T solutions appear to be (0, 6) and (4, 10).
y = -x 2 + 5x + 6
y = −x2 + 5x + 6
y=x+6
6 and 10
Graph
the equations. Find their intersections.
G
Check
{
6 (0, 6)
x
⫺2
O 2
4
The y-values of the solutions are 6 and 10, choice D.
Problem
bl
2
Solving a Linear-Quadratic System Using Substitution
What is the solution of the system of equations? e
Substitute x + 3 for y in
the quadratic equation.
x + 3 = −x 2 − x + 6
Write in standard form.
Factor. Solve for x.
Substitute for x in
y = x + 3.
214
y = −x2 − x + 6
y=x+3
x 2 + 2x − 3 = 0
(x − 1)(x + 3) = 0
x = 1 or x = −3
x=1 S y=1+3=4
x = −3 S y = −3 + 3 = 0
The solutions are (1, 4) and ( −3, 0).
Lesson 5-11 Systems of Linear and Quadratic Equations
8
Problem 3
P
TEKS Process Standard (1)(C)
Solving a Quadratic System of Equations
Which variable should
you substitute for?
You can substitute for
either variable, but
substituting for y results
in a simple equation.
What is the solution of the system? e
W
y = −x2 − x + 12
y = x2 + 7x + 12
M
Method
1 Use substitution.
Substitute y = -x 2 - x + 12 for y in the second equation. Solve for x.
-x 2 - x + 12 = x 2 + 7x + 12
-2x 2
Substitute for y.
- 8x = 0
Write in standard form.
-2x(x + 4) = 0
Factor.
x = 0 or x = -4
Solve for x.
Substitute each value of x into either equation. Solve for y.
y = x 2 + 7x + 12
y = x 2 + 7x + 12
y = (0)2 + 7(0) + 12
y = ( -4)2 + 7( -4) + 12
y = 0 + 0 + 12 = 12
y = 16 - 28 + 12 = 0
The solutions are (0, 12) and ( -4, 0).
Method 2
Graph the equations.
Use a graphing calculator. Define functions Y1 and Y2 .
Plot1
Plot2
\Y1 = –X 2–X+12
2
\Y2 = X +7X+12
\Y3 =
\Y4 =
\Y5 =
\Y6 =
\Y7 =
Plot3
Use the INTERSECT feature to find the points of intersection.
Intersection
X=–4
Y=0
Intersection
X=0
Y=12
The solutions are ( -4, 0) and (0, 12).
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Problem 4
P
TEKS Process Standard (1)(B)
Formulating a Linear-Quadratic System
A Foucault pendulum is used to show the rotation
of the Earth. The pendulum swings along a line
segment and as the Earth rotates, the endpoints
of the segment trace a circle at the base of the
pendulum. Though it is not the most precise
method of telling time, it can be used to tell the
hour of the day with some degree of accuracy.
If the center of the circle is at (0, 0) and the radius
is 13 units, where does the pendulum intersect
the edge of the base when its path (which must
pass through the origin) has slope - 12
5?
The base is circular and is modeled
by a circle with center (0, 0) and
radius 13. The swinging of the
pendulum at a particular time
passes through (0, 0) and has
slope - 12
5.
The location(s) where
the path of the
pendulum intersects the
edge of the base
Write and solve a system of
equations representing the
base and path of the pendulum.
Evaluate the reasonableness of
the solution(s).
The equation of a circle with center (0, 0) and radius 13 is x2 + y 2 = 169. The line
12
passing through (0, 0) with slope - 12
5 is y = - 5 x.
Solve the system e
Why is substitution
a good choice for
solving the system?
One of the equations
in the system is already
solved for y, so you can
save a step if you use
substitution.
Step 1
S
x2 + y 2 = 169
using substitution.
12
y= -5x
Substitute - 12
5 x into the first
equation for y.
(
x2 + - 12
5x
)
2
= 169
144
x2 + 25 x2 = 169
169 2
25 x = 169
Step 2
Substitute 5 and -5 into the
first equation for x.
x2 + y 2 = 169
( {5)2 + y 2 = 169
y 2 = 144
y = {12
x2 = 25
x = {5
The possible solutions are (5, 12), (5, - 12), ( - 5, 12), and ( - 5, - 12).
A line and a circle can intersect in at most two points, so two of the possible
solutions are extraneous. The solutions (5, 12) and ( - 5, - 12) are not solutions
because they do not solve the equation y = - 12
5 x. The solutions to the system
are (5, - 12) and ( - 5, 12).
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Lesson 5-11 Systems of Linear and Quadratic Equations
HO
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RK
O
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PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Solve each system by graphing. Check your answers.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
1. e
y = -x 2 + 2x + 1
y = 2x + 1
2. e
y = x 2 - 2x + 1
y = 2x + 1
3. e
y = x2 - x + 3
y = -2x + 5
4. e
y = 2x 2 + 3x + 1
y = -2x + 1
5. e
y = -x 2 - 3x + 2
y=x+6
6. e
y = -x 2 - 2x - 2
y=x-4
9. e
y = -x 2 + x - 1
y = -x - 1
Solve each system by substitution. Check your answers.
7. e
y = x 2 + 4x + 1
y=x+1
8. e
10. e
y = 2x 2 - 3x - 1
y=x-3
11. e
y = x 2 - 3x - 20
y = -x - 5
12. e
y = -x 2 - 5x - 1
y=x+2
y = -x 2 + 2x + 10
y=x+4
Solve each system.
13. e
y = x 2 + 5x + 1
y = x 2 + 2x + 1
14. e
y = x 2 - 2x - 1
y = -x 2 - 2x - 1
15. e
y = -x 2 - 3x - 2
y = x 2 + 3x + 2
16. e
y = -x 2 - x - 3
y = 2x 2 - 2x - 3
17. e
y = -3x 2 - x + 2
y = x 2 + 2x + 1
18. e
y = x 2 + 2x + 1
y = x 2 + 2x - 1
19. Apply Mathematics (1)(A) A manufacturer is making cardboard boxes by cutting
out four equal squares from the corners of a rectangular piece of cardboard and
then folding the remaining part into a box. The length of the cardboard piece is
1 in. longer than its width. The manufacturer can cut out either 3 * 3 in. squares,
or 4 * 4 in. squares. Find the dimensions of the cardboard for which the volume
of the boxes produced by both methods will be the same.
20. Justify Mathematical Arguments (1)(G) Can you solve the system of equations
x = y 2 + 2y + 1
e
shown by graphing? Justify your answer. Can you solve this system using another y = x - 4
method? If so, solve the system and explain why you chose that method.
Solve each system by substitution.
21. e
x+y=3
y = x 2 - 8x - 9
x+y-2=0
24. e 2
x +y-8=0
22. e
y - 2x = x + 5
y + 1 = x2 + 5x + 3
x2 - y = x + 4
25. e
x-1=y+3
1
23. e
y - 2 x 2 = 1 + 3x
y + 1 x2 = x
2
2y = y - x2 + 1
26. e
y = x2 - 5x - 2
27. Evaluate Reasonableness (1)(B) Your friend wants to solve the system y = x2
and y = x. She concludes that the solution is (0, 0), because 0 = 02 . What would
you tell your friend about her solution?
28. Create Representations to Communicate Mathematical Ideas (1)(E) A circle
with radius of 5 and center at (0, 0) and a line with slope - 1 and y-intercept 7 are
graphed on a coordinate plane. What are the solutions to the system of equations?
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29. Evaluate Reasonableness (1)(B) A system consists of a linear equation and a
quadratic equation whose graph is a parabola. Jason says the solutions of the
system are (0, 0), (2, 4), and ( - 2, 4). Are Jason’s solutions reasonable? Explain.
30. Use a Problem-Solving Model (1)(B) A water taxi travels around an island in a
path that can be modeled by the equation y = 0.5(x - 10)2 . A water skier is skiing
along a path that begins at the point (6, 5) and ends at the point (8, - 4).
a. Write a system of equations to model the problem.
b. Is it possible that the water skier could collide with the taxi? Explain.
Solve each system.
31. e
y = 3x 2 - 2x - 1
y=x-1
34. e
y = 2 x 2 + 4x + 4
1
y = -4x + 122
32. e
y = -x 2 + x - 5
y=x-5
35. e
y = - 4 x 2 - 4x
y = 3x + 8
33. e
y = x 2 - 3x - 2
y = 4x + 28
36. e
y = -4x2 + x + 1
y=x-4
3
1
1
37. Apply Mathematics (1)(A) A company’s weekly revenue R is given by the formula
R = -p2 + 30p, where p is the price of the company’s product. The company is
considering hiring a distributor, which will cost the company 4p + 25 per week.
a. Use a system of equations to find the values of the price p for which the product
will still remain profitable if they hire this distributor.
b. Which value of p will maximize the profit after including the distributor cost?
Determine whether the following systems always, sometimes, or never have
solutions. (Assume that different letters refer to unequal constants.) Explain.
38. e
y = x2 + c
y = x2 + d
39. e
y = ax 2 + c
y = bx 2 + c
40. e
y = (x + a)2
y = (x + b)2
41. e
y = a(x + m)2 + c
y = b(x + n)2 + d
TEXAS Test Practice
T
y = - 14 x 2 - 2x
3
y = x2 + 4
C. 2
42. How many solutions does the system have? e
A. 0
B. 1
D. 3
43. Which expression is equivalent to ( -3 + 2i)(2 - 3i)?
F. 13i
G. 12
H. 12 + 13i
44. Which expression is equivalent to (2 - 7i) ,
7
1
1
7
7
1
A. 8 - 4 i
B. 4 - 8 i
C. 8 + 4 i
2
45. Solve the equation -3x + 5x + 4 = 0. Show your work.
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Lesson 5-11 Systems of Linear and Quadratic Equations
J. -12
(2i)3 ?
1
7
D. 4 + 8 i