Transcript Sections 6.1-6.2

Systems of Equations: Substitution
A system of equations refers to multiple
equations involving multiple variables
 2xy5

3x2y4
A solution is an ordered pair that satisfies
both equations.
(2,1) is the solution to the system
One method for solving a system of
equations is substitution
• Solve for a variable in one equation, and
then plug into the other equation and solve
• Find the value of the other variable by
using one of the original equations
x y 4
Ex. Solve the system 
xy 2
Ex. A total of $12,000 is invested in two funds paying 5%
and 3% simple interest [I = prt]. If the yearly interest
is $500, how much was invested at each rate?
2

x
4
xy
7
Ex. Solve the system 
xy

1
 2
xy4
Ex. Solve the system  2
x y3
One solution: graphs intersect once
Two solutions: graphs intersect twice
No solutions: graphs don't intersect
x2xy1
Ex. Solve the system 
by
 xy1
graphing
Ex. A shoe company invests $300,000 in equipment to
produce a line of shoes. Each pair costs $5 to produce
and is sold for $60. How many pairs of shoes must be
sold before the business breaks even?
Ex. The weekly ticket sales, S, in millions, for a comedy
movie are modeled by the equation S = 60 – 8x, and
the sales model for a drama is S = 10 + 4.5x, where x
is number of weeks that the movie plays. After how
many weeks will the two movie sales be equal?
Practice Problems
Section 6.1
Problems 5, 19, 25, 27, 35, 63
Systems of Equations: Elimination
We are allowed to add the two equations
in a system
By manipulating the coefficients, we can
eliminate a variable and make the system
easier to solve
3x2y4
Ex. Solve the system 
5x2y8
Method of Elimination
• Using multiplication, get the
coefficients of a variable to be
opposites
• Add the equations to eliminate a
variable
• Solve for one variable, then use one of
the original equations to find the other
variable.
7
2x3y
Ex. Solve the system 
5
3xy
5x3y9
Ex. Solve the system 
2x4y14
 2xy1
Ex. Solve the system 
4x2y2
These graphs coincide
 x2y3
Ex. Solve the system 
2x4y1

These graphs are parallel
0
.
0
2
x

0
.
0
5
y


0
.
3
8

Ex. Solve the system 
0
.
0
3
x

0
.
0
4
y

1
.
0
4

Ex. A plane flies into a headwind while making a 2000
mile flight, which takes 4 hr 24 min. If the return
flight takes 4 hr, find the plane's airspeed and the
speed of the wind.
Practice Problems
Section 6.2
Problems 11, 13, 15, 19, 23, 25, 43