Download Solving Systems of Equations Algebraically

Solving Systems of Equations
Algebraically
ACT WARM-UP
 Which
statement best describes the relationship
between the graphs of y = 2 and x = 2?
 A) The two lines have the same slope.
 B) The lines are perpendicular.
 C) The lines are parallel.
 D) The lines intersect at (2, 0).
 E) None of the above.
 y = 2 is a horizontal line and x = 2 is a vertical
line. Therefore, the answer is B) perpendicular.
Objectives
 Solve
systems of equations by substitution.
 Solve systems of equations by elimination.
Essential Question
How do you solve a system of linear equations
by substitution?
Substitution Method
The substitution method is used
to eliminate one of the variables
by replacement when solving a
system of equations.
Think of it as "grabbing" what one
variable equals from one equation
and "plugging" it into the other
equation.
Elimination method
Simultaneous equations written
in standard form got you
baffled? Relax! You can do it!
Think of the adding or combining
like terms method as simply
"eliminating" one of the variables
to make your life easier.
Solving Systems
You will need paper and pencil
or a whiteboard to solve the
following problems.
Use substitution to solve the systems of equations.
x  26  4 y
x  5 y  10
Substitute 26 – 4y for x in the second equation
and solve for y.
Second equation
Substitute 26 – 4y for x.
Subtract 26 from each side.
Divide each side by –9.
Now substitute the value for y in either of the original
equations and solve for x.
x  26  4 y
First equation
x  26  4(4)
Replace y with 4
x  26  16
Simplify
Subtract 16 from each side.
Answer: The solution of the system is (10, 4).
x  3y  2
x  7 y  12
Answer: (5, 1)
Use the elimination method to solve the system
of equations.
In each equation, the coefficient of x is 1. If one equation
is
multiplied by -1, the variable x will be eliminated when the
two equations are added together.
x  2 y  10  x  2 y  10
1 x  y  6    x  y  6
y4
Now find x by substituting 4 for y in either original equation.
Second equation
Replace y with 4.
Subtract 4 from each side.
Answer: The solution is (2, 4).
Use the elimination method to solve the system
of equations.
Answer: (17, –4)
Use the elimination method to solve the system
of equations.
Multiply the first equation by 2 and the second equation
by 3. Then add the equations to eliminate the y variable.
Multiply by 2.
Multiply by 3.
Replace x with 3 and solve for y.
First equation
Replace x with 3.
Multiply.
Subtract 6 from each side.
Divide each side by 3.
Answer: The solution is (3, 2).
Use the elimination method to solve the system
of equations.
Answer: (–5, 4)
Use the elimination method to solve the system
of equations.
Use multiplication to eliminate x.
Multiply by 2.
Answer: Since there are no values of x and y that will
make the equation
true, there are
no solutions for the system of equations.
Use the elimination method to solve the system
of equations.
Answer: There are no solutions for this system
of equations.
Solve Systems
Solve these
problems
algebraically,
using either
the
substitution or
addition/
combination
method.
a.
A) x + 4y =26, x- 5y = - 10
B) x –3y = 2, x + 7y = 12
C) 3x –y = 7, x + 4y = 11
D) 2x –3y = 11, 2x +2y = 6
E)
x + 2y = 10, x + y = 6
F)
x +3y = 5, x +5y = -3
G) 2x + 3y = 12, 5x – 2y = 11
Skiing trip: 40 members of South Aiken High School
went on a one-day ski trip. They can rent skis for $10
per day or snowboards for $12 per day. All members
paid a total of total $420. Write a system of equations
that represents the number of members who rented
two types of equipment? How many members rented
skis and how many rented snowboards?
Let x = represent skis, y =represent snowboards
# snowboards
plus
y
+
# skis
x
is
=
Members
40
$ Amount
$ cost per day
is
for skis
$ cost per day
plus
for Snowboard.
+
420
=
Answer:
x + y = 40
10x + 12y = 420
30 members rented
skis and 10 members
rented snowboards.
10x
12y
Mr. Talbot is writing a test for his science classes. The
test will have true/false questions worth 2 points each
and multiple-choice questions worth 4 points each for
a total of 100 points. He wants to have twice as many
multiple-choice questions as true/false. Write a system
of equations that represents the number of each type
of question. How many true/false questions and
multiple-choice questions will be on the test?
Answer:
2x +4y = 100
y = 2x
(10(T/F), 20(M/C) )
Three times one number added to five times another
number is 54. The second number is two less than the
first. Find the numbers
Let x = first number, y = the second number
3*#
plus
3x
+
5*other #
is
Total
5y
=
54
Other #
First #
2 less than
is
y
=
x
Answer:
3x + 5y = 54
y=x–2
(8,6)
-
2
The average of two numbers is 7. Find the numbers if
three times one number is one half of the other
number.
Answer:
(x + y)/2 = 7
3x= 1/2y
(2, 12)
Essential Question
How do you solve a system of linear equations
by substitution?
Solve one equation for one variable in terms of the
other. Then, this expression is substituted for the
variable in the other equation.
Solving Systems Decisions
– when a finding a rough solution or
determining if a solution exists
 Substitution – when one or both equations are
given in the slope-intercept form
 Elimination – when the equations are in standard
form. Multiply the equations in order to get
opposite coefficients so variables will cancel when
you add the equations together
 Graphing