Download Hon Alg 2: Unit 6 SYSTEMS OF EQUATIONS A system of equations

Hon Alg 2: Unit 6
SYSTEMS OF EQUATIONS
A system of equations is two (or more) equations with the same variables.
y  5  3x
Example:
5 y  4x  3
The SOLUTION to a system of equations is the ordered pair that satisfies (solves) ALL of the equations.
Which of the following ordered pairs is a solution to the example system?
a. (-2, -11)
b. (2, 1)
c. (1, -2)
There are THREE TYPES OF SOLUTIONS to any system of equations:
1) ONE SOLUTION: Exactly one ordered pair (x, y) will solve all equations.
2) NO SOLUTION: NO ordered pair (x, y) will EVER solve all equations.
3) INFINITELY MANY SOLUTIONS: an infinite number of ordered pairs will solve all equations.
There are THREE SOLVING METHODS for system of equations:
1) GRAPHING
2) SUBSTITUTION
3) ELIMINATION
Method #1: GRAPHING: Find the intersection of lines
STEP 1: Make sure each equation is in slope-intercept form ( y = mx + b)
STEP 2: [Y =], for Y1 write your first equation, for Y2 write your second equation
STEP 3: Adjust [WINDOW] is large enough with XIMN + XMAX and YMIN + YMAX
STEP 4: [2ND]  [CALC (Trace)]  [5:Intersect]  Scroll to intersection
STEP 5: [ENTER] Three Times …
STEP 6: Solution Ordered Pair is Intersection: X = #, Y = #
2 x  5 y  20
EXAMPLE - Step-by-Step: Solve by graphing:
1. Solve for y:
 3x  y  4
2 x  5 y  20
Equation #1:  5 y  2 x  20
2
y  x4
5
 3x  y  4
Equation #2:
2. GRAPH
3. Find INTERSECTION (use calculator steps)
4. SOLUTION: (-3.08, -5.23)
y  3x  4
More Key Terms about Solutions of Systems of Linear Equations:
CONSISTENT: The graphs of each equation ____________________________.
There is ___________________________________________ ordered pair that satisfies both equations.

INDEPENDENT System: the system has ____________________________________ solution.
o The two lines intersect at ______________________________

DEPENDENT System: the system has an __________________________________ of solutions.
o The two lines are the _________________________.
INCONSISTENT: The graphs of each equation are ____________________________.
There are __________ ordered pairs that satisfy both equations.
SUMMARY TABLE:
Intersecting Lines
Same Line
y
Parallel Lines
y
Graph of a
System
x
y
x
x
Number of
Solutions
Terminology
GRAPHING - Examples: Make sure you have the equations in slope intercept - form!!!
x y7
y  3x  5
(1)
y  4x  3
(2)
y  3 x  12
y  2x  1
(4)
y  x  7
x y3
x  2y  5
(3)
(5)
6 x  4 y  14
2x  y  4
3 x  5 y  27
(6)
x  10 y  23
Special Case – VERTICAL LINE:
Vertical lines cannot be graphed in calculator because equation is x = number.
 SUBSTITUTE x-value of vertical line into other equation to solve
3x  2 y  4
9  4x  3 y
x4
(8)
(9)
(7)
x  1
x4
y  5 x  8
Method #2: SUBSTITUTION - substitute one equation into the other
-5x = 3y – 4
y + 2x = 6
3y = 2x + 11
y+x=7
STEP 1:
SOLVE one of the equations for the given variable (y = … or x = … )
Solve y + x = 7 for x.
x=7–y
STEP 2:
SUBSTITUTE with parentheses the expression for x or y (STEP 1) into the other equation
3y = x + 13
Substitute: (7 – y) for x
*watch the parentheses*
3y =2 (7-y) + 11
STEP 3:
SOLVE the equation created in STEP 2
Solve 3y = 2(7 - y) + 11 for y,
3y = 14 – 2y + 11
3y = -2y + 25
5y = 25
y=5
STEP 4:
Use SOLUTION to STEP 3 and find the value of the remaining variable in one of the
original equations.
Substitute y = 5 into y + x = 7
5+x=7
x = 7 -5
x=2
STEP 5:
Write the solution to the system as an ORDERED PAIR (x, y)
x = 2; y = 5
Solution: (2, 5)
EXAMPLE Step-by-Step: Solve
2 x  5 y  20
 3 x  y  4 by substitution method.
Solve for y in second equation
 3x  y  4
 y  3x  4
Substitute second equation into first for the variable y
2 x  5 y  20
 2 x  5( 3 x  4)  20
 40 68 
,  or (-3.08, -5.23) …
 13 13 
Solution  
Solve first equation
2 x  15 x  20  20
 13 x  40

x
40
13
Substitute solution into second equation
x
40
68
 40 
 y  3    4  y  
13
13
 13 
PRACTICE SOLVE BY SUBSTITUTION:
Hint: It doesn’t matter which equation you choose or which variable you decide
to solve for as your answer should be the same.
5 x  4 y  22
2 x  y  11
x  4y  5
(3) y  10  2 x
(1) x  y  13
(2) 3 x  2 y  34
2x  3 y  9
(4)
5 x  3 y  17
x  2y  8
(5) 1 x  y  18
2
Method #3: ELIMINATION - eliminate one variable from both equations
STEP #1: Put both equations in SAME FORM (Ax + By = C)
STEP #2: SELECT A VARIABLE to eliminate (x or y)
STEP #3: As needed, make that variable in both equations have same coefficients and
opposite signs (ex: -3x and 3x) by MULTIPLICATION to each equation. (LCM can help)
STEP #4: ADD the two equations together to eliminate that variable term
STEP #5: Solve the resulting one-variable equation
STEP #6: Find the value of the eliminated variable by substituting previous solution into
one of the given equations
STEP #7: Write solution as an ordered pair
Example of Step-by-Step: ELIMINATION METHOD
2 x  5 y  20
Solve  3 x  y  4 by elimination
1. Both equations already in same form (roughly standard)
2. Eliminate y: y-terms are -5y and 1y LCM of 5 and 1 = 5
3. Multiply second only equation by 5
4. ADD Equations
2 x  5 y  20
 15 x  5 y  20
 13 x  40
2 x  5 y  20
5( 3 x  y )  5(4)
5. Solve equation
 13 x  40
40
x
13
 Now System:
2 x  5 y  20
 15 x  5 y  20
6. Substitute solution back
into one of the original
 40 
x     3 x  y  4
 13 
68
 40 
 3    y  4  y  
13
 13 
 40 68 
7. Solution:   ,  or (-3.08, -5.23)
 13 13 
PRACTICE SOLVE BY ELIMINATION:
2 x  7 y  19
3 x  5 y  12
(2) 5 x  2 y  67
(1) 6 x  2 y  0
4a  2b  15
(4)
2a  2b  7
3 x  7 y  14
(5) 5 x  2 y  45
x  5 y  121
(3) 6 x  2 y  34
3 x  4 y  8
(6) 5 x  6 y  27
Hon Alg 2: Unit 6
Name: _________________________________
SOLVING SYSTEMS OF EQUATIONS GENERAL PRACTICE
While GRAPHING, SUBSTITUTION, and ELIMINATION Methods will find the same solution to a given
system of equations, each one has advantages and disadvantages based on the initial set up of the system.
I. Solve each system of equations by GRAPHING
 Show y-intercept form of your equations
y  3x  8
y  5x  3
1)
6x  2 y  4
2)
 4x  3 y  6
y  11
3)
II. Solve each system of equations by SUBSTITUTION
x  2y  5
4)
3x  4 y  7
3x  y  2
x  2y  8
5)
2x  3 y  2
6)
III. Solve each system of equations by ELIMINATION
2x  y  2
7)
4 x  9 y  3
x  3 y  13
8)
3x  5 y  5
11)
 2x  7  4 y
5x  y  6
 y  3 x  1
9)
5x  1  3 y
2x  3 y  3
10)
5x  y  1
7 x  5 y  20
3 x  3 y  39
3 x  4 y  25
12)
 7 y  4 x  16
SYSTEM OF EQUATION WORD PROBLEMS
1.
2.
3.
4.
Read the problem Carefully and Identify Any Important Information
Define Variables (x and y)
Write a system (2 equations) with variables
Solution Method: Based on your equations, decide on the best method
(Graph, Substitution, Elimination)
5. Answer the QUESTION
1)
The sum of two numbers is 200 and their difference is 28.What are the two numbers?
2)
100 cans and 300 bottles were bought for $450. 200 cans and 200 bottles were bought for $400.
What is the cost of an individual can and bottle?
3)
Arsene saves $55 per week and already has saved $270. Jose has $990 and spends $35 per week.
How many DAYS until Arsene and Jose have the same amount of money?
4)
There are a total of 800 students in a high school. Twice the number of upperclassmen is 98 less
than the number of underclassmen. How many upper and underclassmen are in the high school?
5)
A total of 10,000 king and queen sized sheets were sold by a warehouse. A king sized sheet costs
$25 and a queen sized sheet costs $20. All 10,000 sheets were sold for a total of $227,000. How
many queen sheets were sold?
6)
The width of a rectangle is 5 times larger than the length of the rectangle. The
perimeter of the rectangle is 360 units. What are the measurements of the length
and width?
7)
The length of a rectangle is 10 more than twice the width. The perimeter is 70 units.
What is the area of the rectangle?
8)
Stephan has been gaining weight at 1.5 pounds a week from his original weight of
120 pounds. Seth is losing weight at 2.5 pounds a week from his original weight of
184 pounds. How much will Seth and Stephan weigh when they are the same
weight?
9)
Georgia bought 3 pens and 4 notebooks for $26.50 from the bookstore. She later
bought 7 pens and 5 notebooks for $41.25 from the same bookstore. How much
would you have to pay for 2 pens and 2 notebooks?
10) You are able to buy 4 tacos and 6 burritos for $62.58 or you could buy 5 burritos
and 7 tacos for $66.78. How much would 3 burritos cost?
11) Two numbers have a sum of 117. The larger number is 42 less than twice the
smaller number. What are the numbers?
12) Nancy has 113 coins that are all dimes and quarters. The value of the coins is exactly
$20.00. How many dimes and quarters does Nancy have?
13) Two numbers have the difference of 38. The larger number is 5 more than
quadruple the smaller. What are the numbers?
14) Mr. Grimstead is writing a test for his Algebra 2 class. It will have multiple choice
questions that are worth 4 points each and true/false questions that are worth 3
points each. The total number of points for the test is 100. He wants the number of
true/false questions to be ¾ the number of multiple choice questions. How many of
each type of question are there on the test?
15) A boat travels with the current and can cover 100 miles in 4 hours. If the boat
travels against the current, it can only travel 80 miles in the same time. How fast is
the boat in still water, and how fast is the current.