Download Using the Calculator to Solve a System of Equations Graphically

Day 1 - Solving Systems of Equations
Algebra 1A
Unit 7 – Systems of Equations and Inequalities
Warm-Up:
Write the equation of the line associated with each graph.
1.
2.
3.
m = _____ b = _______
m = _____ b = _______
____________________
____________________
Does the point (-1, 5) lie on the line
2y + x = 9? Justify your answer.
REVIEW:
What best describes the relationship between the graphs of the equations, y = 2x – 3 and y  
(a) The graphs are parallel
1
x 1?
2
Slopes are ________
Lines intersect at __________ point(s)
(b) The graphs are the same line
Slopes are ________,
Y-intercepts are ___________,
Lines intersect at __________ point(s)
(c) The graphs are perpendicular
Slopes are _________________ ____________________
Lines intersect at __________ point(s)
(d) None of the above
When graphing two lines on the same set of axes there are three types of solutions:
One Point
No Solution
(parallel lines)
Infinitely Many Solutions
(same line)
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Solving Systems of Equations Graphically
Graph more than one equation on the same set of axes.
The SOLUTION of the system of equations
is the POINT they share in common.
This is where the lines INTERSECT.
STEPS TO TAKE:
1.
2.
3.
4.
5.
Put each equation in y = mx + b form.
State slope (m) and y-intercept (b) for each equation.
Graph and label each equation.
Find the point where the lines intersect (POI). Label it with its ordered pair.
THIS is YOUR SOLUTION.
Check point in both equations of the lines. A check box may be useful.
Examples: Solve the following systems of equations graphically and check.
#1.
y  2x  1
y x  4
m = _____
m = _____
b = ______
b = _______
Point of Intersection (POI): _______
CHECK:
#2.
PARALLEL / PERPENDICULAR / SAME / OTHER
y2
y  2x  5
m = _____
m = _____
b = ______
b = _______
Point of Intersection (POI): _______
CHECK:
2
1
x
3
PARALLEL / PERPENDICULAR / SAME / OTHER
Examples: Solve the following systems of equations graphically and check.
3x  y  5
#3.
y
1
x5
3
m = _____
m = _____
b = ______
b = _______
Point of Intersection (POI): _______
CHECK:
#4.
PARALLEL / PERPENDICULAR / SAME / OTHER
y x2
1  y  2x
PARALLEL / PERPENDICULAR / SAME / OTHER
#5.
y  2
x  5
PARALLEL / PERPENDICULAR / SAME / OTHER
In general:
____________ and _____________ lines are always _______________ to each other and
the point of intersection is (x, y) 
(
,
)
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#6.
1. What is the relationship between the lines y = 2x – 3 and y – 2x = 1?
_______________________________________
2. Graph the following system of equations.
3. What point(s) do they both share in common? _______
4. How many places do the graphs intersect? _______
5. What is the solution to the systems of equations?
_____________________________________
Lines that are _________________ have _______ solution.
#7.
1. What is the relationship between the lines y = x – 3 and y – x + 1 = -2?
_______________________________________
2. Graph the following system of equations.
3. What point(s) do they both share in common? _______
4. How many places do the graphs intersect? _______
5. What is the solution to the systems of equations?
_____________________________________
Lines that are _________________ have _____________ solutions.
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DAY 1 HOMEWORK
Solve the System of Equations by Graphing
1.
y = 2x
y = -2x + 4
Solution: _________
2.
y=5
x=2
Solution: _________
3.
y+4=x
PARALLEL / PERPENDICULAR / SAME / OTHER
PARALLEL / PERPENDICULAR / SAME / OTHER
y = -x
Solution: _________
PARALLEL / PERPENDICULAR / SAME / OTHER
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Solve the System of Equations by Graphing
4.
y = 4x + 5
-4x + y = -3
Solution: _________
5.
2x – y = -5
-2x – y = -1
Solution: _________
6.
y=½x+1
PARALLEL / PERPENDICULAR / SAME / OTHER
2y – x = 2
Solution: _________
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PARALLEL / PERPENDICULAR / SAME / OTHER
PARALLEL / PERPENDICULAR / SAME / OTHER
Day 2 - Solving Systems of Equations
Algebra 1A
Unit 7 – Systems of Equations and Inequalities
Warm-Up:
1. Is the point (2, -2) the solution of the system x + y = 0 and 2x – y = 6? Justify your answer.
2. Is the point (5, -3) the solution of the system y = x - 8 and 2x – y = 7? Justify your answer.
QUADRANTS:
Numbered sections of the coordinate plane.
II
I
III
IV
The point lies in which quadrant
a)
(5, -2)
_______
e)
(5.2, 7.4) _______
b)
(4, 6)
_______
f)
(-1, -1/2)
_______
c)
(-6, -1)
_______
g)
(-3, 3.3)
_______
d)
(-1, 3)
_______
h)
(1/3, -5/7) _______
In general:
Point in Quadrant I
(
,
)
Point in Quadrant III
(
,
)
Point in Quadrant II
(
,
)
Point in Quadrant IV
(
,
)
Graph the following systems of equations, state the solution and the quadrant where the
solution lies.
1.
x = -3
y=4
Solution: ________ Quadrant: ____
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2.
y
5
x3
3
y
1
x3
3
Solution: ________ Quadrant: ____
3.
y – 3x = – 4
y
1
x3
2
Solution: ________ Quadrant: ____
4.
y – 3 = 4x
x + y = -2
Solution: ________ Quadrant: ____
5.
y = 2x – 3
y – 2x = 1
Solution: ________ Quadrant: ____
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DAY 2 HOMEWORK
Graph the following systems of equations, state the solution and the quadrant where the
solution lies.
1.
y = -2
x=5
Solution: ________ Quadrant: ____
2.
x+y=4
2x – y = 2
Solution: ________ Quadrant: ____
3.
2y = -x – 4
-2y = 3x - 4
Solution: ________ Quadrant: ____
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4.
y + 4 = -2x
x = -2
Solution: ________ Quadrant: ____
5.
y=½x–3
-4y = x
Solution: ________ Quadrant: ____
6.
y = 3x – 2
y–2=x
Solution: ________ Quadrant: ____
7. Can there be more than one point of intersection between the graphs of two linear
equations? Why or why not?
__________________________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
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Day 3a - Solving Systems of Equations
Calculator/Word Problems
Algebra 1A
Unit 7 – Systems of Equations and Inequalities
Warm-up
State the quadrant in which the point lies.
a)
(.5, 2) _______ b) (4, -3.6) _______
c) (-1/6, -1)
_______ d) (-1,
)
_______
Without graphing, determine how many solutions there are for the systems of equations?
1.
y = 1/2x – 2
and y = -2x + 5
(a) 0
(b) 1
(c) 2
(d) infinitely many
2.
y = 3x -1 and y = 3x + 8
(a) 0
(b) 1
(c) 2
(d) infinitely many
3.
y = 2x + 1 and 3y = 6x + 3
(a) 0
(b) 1
(c) 2
(d) infinitely many
Using the Calculator to Solve a System of Equations Graphically
1.
2.
3.
4.
Put each equation into y = mx + b form. (Solve for y)
Enter these equations as y1 = and y 2 =
GRAPH
Find the point of intersection one of two ways:
a. 2nd TRACE(CALC), 5: intersect, enter, enter, enter
b. 2nd GRAPH(TABLE)  the x value that has the same y1 and y 2 values
Let us check our homework answers using the calculator.
Solving a System of Equations WORD PROBLEMS 
a)
Use LET statements and TWO variables (independent (x), dependent (y))
b) Create TWO equations using the variables in the LET statements
c) Solve equations for the dependent variable (y)
d) Enter equations on calculator (y =)
e) Adjust WINDOW and GRAPH (Sketch the graph)
e) Find the point of intersection (2nd, TRACE, 5:INTERSECT)
f)
Sentence Answer!!!
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Solve the system of equation word problems by using the box method.
1.
The difference of two numbers is 3 and their sum is 13. Find the numbers.
SENTENCE: _________________________________________________________
2.
The sum of the distances two hikers walked is 53 miles, and the difference is 25 miles.
What are the distances hiked?
SENTENCE: _________________________________________________________
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3.
Katie’s bank holds only $5 and $10 bills. There are a total of 26 bills in all. The bank
holds $170. Find the number of $5 and $10 bills that Katie has.
SENTENCE: _________________________________________________________
4.
Michael has $5.70 in a jar. The jar only holds nickels and quarters. All together there
are 34 coins in the jar. Find the number of nickels and quarters that Michael has.
SENTENCE: _________________________________________________________
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DAY 3a HOMEWORK
1.
The sum of two numbers is 29 and the difference of the numbers is 11. Find the
numbers.
SENTENCE: _________________________________________________________
2.
Jocelyn has $1.95 in her pocket made up of 27 nickels and dimes. How many of each
type of coin does she have?
SENTENCE: _________________________________________________________
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3.
A student bought 1 box of crayons and 5 reams of paper for $54. She bought 5 boxes
of crayons and 3 reams of paper for $50. What is the cost of each box of crayons and
each ream of paper?
SENTENCE: _________________________________________________________
4.
A shopper purchased 4 tables and 2 chairs for $200 and 2 tables and 7 chairs for $400.
What is the cost of each table and each chair?
SENTENCE: _________________________________________________________
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Day 3b - Solving Systems of Equations
Calculator/Word Problems
1.
Algebra 1A
Unit 7 – Systems of Equations and Inequalities
A farm raises a total of 220 chickens and pigs. The number of legs of the stock in the
farm totals 520. How many chickens and pigs are at the farm?
SENTENCE: _________________________________________________________
2.
Suppose you bought 8 mangoes and 3 apples for $18 and 3 mangoes and 5 apples for
$14.50. How much does each mango and each apple cost?
SENTENCE: _________________________________________________________
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3.
The school that Stefan goes to is selling tickets to a choral performance. On the first
day of ticket sales the school sold 3 adult tickets and 1 child ticket for a total of $38.
The school took in $52 on the second day by selling 3 adult tickets and 2 child tickets.
Find the price of an adult ticket and the price of a child ticket.
SENTENCE: _________________________________________________________
4.
One electrician charges a $50 service call fee plus $25 per hour for labor. Another
electrician charges a $35 service call fee plus $27.50 per hour for labor. How long
would a job take if the total cost for each electrician is the same?
SENTENCE: _________________________________________________________
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5.
A cell phone provider offers a plan that costs $40 per month plus $.20 per text message
sent or received. A comparable plan costs $60 per month but offers unlimited text
messaging.
a) How many text messages would you have to send or receive in order for the
plans to cost the same each month?
SENTENCE: _________________________________________________________
b) If you send or receive an average of 50 text messages each month, which plan
would you choose? Why?
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DAY 3b HOMEWORK
1.
At a local fitness center, members pay a $20 membership fee and $3 for each aerobics
class. Nonmembers pay $5 for each aerobics class. For what number of aerobics
classes will the cost for members and nonmembers be the same?
SENTENCE: _________________________________________________________
2.
You are looking for a job. One job pays $9 per hour. Another pays $12 per hour, but
you must buy a uniform that costs $39. After how many hours of work would your net
earnings from either job be the same?
SENTENCE: _________________________________________________________
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3.
Your math test has 38 questions and is worth 200 points. The test consists of multiplechoice questions worth 4 points each and open-ended questions worth 20 points each.
How many of each type of question are there?
SENTENCE: _________________________________________________________
4.
A plant nursery is growing a tree that is 3 ft. tall and grows at an average rate of 1 ft.
per year. Another tree at the nursery is 4 ft. tall and grows at an average rate of 0.5
ft. per year. After how many years will the trees be the same height?
SENTENCE: _________________________________________________________
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Day 4 – Graphing Inequalities on a Set of Axes
Algebra 1A
Unit 7 – Systems of Equations and Inequalities
Warm-up:
Solve for x, graph the inequalities on the number line and write in interval notation.
2x  3  15
5  3x  1  8
Solve for y. Put each inequality into y “=” mx + b form.
REMEMBER: When multiplying or dividing by a negative  FLIP the symbol!
1.
x+y<7
2.
3.
2x + 3y > 6
4.
-2y - 9 ≤ 4x
3+y≥x
How do these problems differ from those in the warm-up?
Can these inequalities be graphed on a number line?
Find a point (x, y) that would satisfy the inequality in problem #1.
Is the point (-5, 7) a solution of the linear inequality?
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Graphing Inequalities on the Coordinate Plane
Ex. y < 2x + 3
Just like graphing linear equations with 2 extra steps.
1.
Must draw the line either
dotted (if < or >)
2.
Must shade.
Shade above (if > or ≥) when in y=mx + b form.
Shade below (if < or ≤) when in y=mx + b form.
or
solid (if ≤ or ≥).
3. CHECK: Test a point in the shaded region.
1.
2x + 3y > 6
m=
22
2.
b=
4x – y < 2
m=
b=
line: DOTTED / SOLID
line: DOTTED / SOLID
shade: ABOVE / BELOW
shade: ABOVE / BELOW
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Day 5 - Solving Systems of Inequalities
Algebra 1A
Unit 7 – Systems of Equations and Inequalities
Warm-Up: Write the linear inequality represented by the following graphs?
m=
b=
dotted/solid
shade above/below
INEQUALITY: _________
m=
b=
dotted/solid
shade above/below
INEQUALITY: ________
Solving Systems of Linear Inequalities
1. Put both inequalities in y = mx + b form.
Identify m, b, "dotted or solid" and "shade above or below" for both.
2. Graph both inequalities. Including dotted/solid line and shading.
LABEL both of them.
3. The solution to the system is the entire area of the graph that is shaded by both
inequalities. (The plaid part.)
4. Label the "Solution Set" with the letter "S" or another letter you are
instructed to use.
5. Check: Use a point in the solution set to check both inequalities.
Solve the system of inequalities:
y < 2x + 1
x + y ≥ -1
m=
y < 2x + 1
b=
dotted or solid
Shade above or below?
m=
x + y ≥ -1
b=
dotted or solid
Shade above or below?
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Solve the system of inequalities:
y  3
2x  y  6
m=
b=
dotted or solid
Shade above or below?
m=
b=
dotted or solid
Shade above or below?
Solve the system of inequalities:
m=
b=
dotted or solid
Shade above or below?
m=
x - y ≥ -2
b=
dotted or solid
Shade above or below?
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x4
x - y ≥ -2
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Day 6 - Solving Systems of Inequalities Word Problems
Algebra 1A
Unit 7 – Systems of Equations and Inequalities
Warm-Up:
Suppose you are graphing a system of two linear inequalities, and the boundary lines are parallel. Does that
mean that the system has no solution?
Got It? What system of inequalities is represented by the graph?
a. Find the inequality shown by the dashed line.
Determine the equation of the dashed line.
The shaded region is above / below the line.
The boundary line is solid / dashed .
What inequality symbol should you use to
write the inequality for this graph?
b. Find the inequality shown by the solid line.
Determine the equation of the solid line.
The shaded region is above / below the line.
The boundary line is solid / dashed .
What inequality symbol should you use to
write the inequality for this graph?
1. EXAMPLE: You want to build a fence for a rectangular dog run. You want the run to be at least 10 ft wide. The run
can be at most 50 ft long. You have 126 ft of fencing. What is a graph showing the possible dimensions of the dog run?
Remember, a dog run must have fencing on all four sides. Follow the steps to write a system of inequalities and
graph it.
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Define what each variable represents.
Let x = the width of the dog run.
Let y =
Write a system of inequalities.
2x + 2y
… this is y
x
y
Graph the system of inequalities.
Why is the graph only in the first quadrant?
_______________________________________________________________________
_______________________________________________________________________
Use the graph to write possible dimensions of the dog run. width (x): length (y):
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2. FRUIT:
Cherries cost $4 per pound. Grapes cost $2.50 per pound. You can spend no more than $15 on fruit, and you need at
least 4 pounds in all. Draw a graph showing the amount of each fruit you can buy?
3. TIME MANAGEMENT:
You are planning what to do after school. You can spend at most 6 hours daily playing basketball and doing homework.
You want to spend less than 2 hours playing basketball. You must spend at least 1.5 hours on homework.
What is the graph showing how you can spend your time?
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4. EARNINGS:
Suppose you have a job mowing lawns that pays $12 per hour. You also have a job at a clothing store that pays $10 per
hour. You need to earn at least $350 per week, but you can work no more than 35 hours per week. You must work a
minimum of 10 hours per week at the clothing store. What is a graph showing how many hours per week you can work at
each job?
5.
BREAD:
Britney wants to bake at most 8 loaves of bread for a bake sale. She wants to make banana bread that sells
for $1.25 each and nut bread that sells for $1.50 each and make at least $12 in sales. Write a system of
inequalities for the given situation and graph the inequalities.
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6.
The owner of an ice cream stand needs to order waffle cones and sugar cones. There is room to store 10
boxes of cones. Each box of sugar cones costs $100, and each box of waffle cones costs $150.
He has $1250 budgeted for the purchase of cones. Write a system of inequalities and graph.
7. For the school fundraiser, a class is selling stationery and greeting cards. The goal for the class is to sell at least 100
items. The school receives $2.50 for each stationery set that is sold and $3 for each set of greeting cards
that is sold. The goal is to raise at least $300. Write a system of inequalities and graph.
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DAY 6 HOMEWORK
1. GARDEN:
You are fencing in a rectangular area for a garden. You have only 150 feet of fence. You want the length of the garden to
be at least 40 feet. You want the width of the garden to be at least 5 feet. Draw a graph
showing all the possible dimensions for your garden?
a) What variables will you use? What will they represent?
b) How many inequalities do you need to write?
2. GIFT CERTIFICATES:
You received a $100 gift certificate to a clothing store. The store sells T-shirts for $15 and dress shirts for $22. You want
to spend no more than the amount of the gift certificate. You hope to buy at least 5 items with your gift. You need
at least one dress shirt. What are all of the possible combinations of T-shirts and dress shirts you could buy?
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3. STUDENT JOBS:
Mark is a student, and he can work for at most 20 hours a week. He needs to earn at least $75 to
cover his weekly expenses. His dog-walking job pays $5 per hour and his job as a car wash
attendant pays $4 per hour. Write a system of inequalities to model the situation, and graph the
inequalities.
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