math-3 lesson 7-4 solving systems of equations...
TRANSCRIPT
Solution of a System
A solution of a system of two equations in two variables is an ordered pair of real numbers that lies on both graphs (where the graphs intersect).
Methods of Solving Systems
3. Graphing: The points of intersection of the graphs
are the solutions.
2. Elimination: another algebraic method that we
learned in Math-2.
1. Substitution: a method using algebra and properties of
algebra (we learned this in Math-2)
Graphing is extremely useful for very difficult equations!
Garden Problem: Find the dimensions of a rectangle with a perimeter of 200 m and an area of 500 m2.
x
y20022 yx
500xy
2
)2200( xy
x
y 500
Rewrite each equation by solving for ‘y’.
Solve by graphing.
Garden Problem: Find the dimensions of a rectangle with a perimeter of 200 m and an area of 500 m2.
x
y
xx
y
1002
2200
xy 500
Solve by graphing. Adjust window.
Oops, window is too small.
Solution: (5.3, 94.7)
Width = 5.3 m
Length = 94.7 m
How would we solve the gold alloy mixture problem?
A 15 carrot (Ct) ring, that weighs 20 gm, is melted down. It
is mixed with the metal from a melted down 12 Ct gold ring
that weighed 30 gm. Pure gold is 24 Ct. What % gold is the
mixture?
30 gm20 gm
625.024
15 5.0
24
12
50 gm
x
mixturein gold ofamount )30(5.0)20(625.0
5.27
xwhole
part
50
5.27%5555.0
How could you turn the following single variable equation into a system of equations in two variables?
Is it possible?
4537 x
453
7
xy
y
turn the following single variable equations into a system of equations in two variables.Your turn:
1045 2 xx
104
5
2
xxy
y
turn the following single variable equations into a system of equations in two variables.
Your turn:
)73log(264 x
)73log(26
4
xy
y
Solving a single variable equation
2)4log(34 x
4y 2)4log(3 xy
We know that:
y = y
left side of the equation
equals right sideSet left side
equal to ySet right side
equal to y
The solution to the two equations will be (x, 4) where
‘x’ is the solution to the single variable equation.
Check: 3log(8.64 – 4) + 2 = 3.99955
A solution has a pH of 2.3. What is the hydrogen-ion concentration? pH = -log [H+]
xy
y
log
3.2
]log[3.2 H
005.0x
]log[3.2 H
65log93 ds
Formula relating distance (d) that a tornado travels
and the wind speed (s) inside the cone of the tornado.
The wind speed of a tornado was 251 mi/hr. How far did it
travel on the ground?
65log93251 d
miles 100d
10
9313
2
x
xx
10
797
xxy
10
93
13
2
x
xxy
y
This has an oblique asymptote
since x + 10 doesn’t divide evenly.
9- 3 110
79- 7- 107 10 -
81.17
81.7
x
x
0
7 xy
Finding the equation of a circle:
What are the (x, y) coordinates of the point?
225 r
321
1
4
2
3
What is the radius
of the circle?
222 cba
222 43 r
2169 r
)4 ,3(
25169
Now I will pick a random point on the circle.
x
222 ryx ) ,( yx
y
What is the equation for
the radius of the circle
when the point is (x, y)?
This is the general
equation for a circle
centered at (0, 0)!!!
x
222 ryx
r = 5222 5 yx
2522 yx
This is the equation of a circle centered at the origin
whose radius is 5.
Graphical TransformationsParent Function: The simplest function in a family
of functions (lines, parabolas, cubic functions, etc.)
2xy
Compare the two parabolas.
Replacing ‘x’ with ‘x – 2’
translates the parent
function right by 2.
2)2( xy
2xy 22 xy
Adding 2 to the parent
function moves the
function up 2 units.
Subtract 2 from both sides yields:22 xy
2)2( xy
Replace “y” in the parent function with (y – 2) moves the
graph up 2.
2xy
Replacing ‘x’ with
(x – 2) moves the
parent function
right by 2.
2)2( xy2)2( xy
Replace “y” with
(y – 2) moves
the parent
function up by 2.
Let’s move the circle 2 spaces to the right.
How do we change the equation to translate the graph
right 2?
x
222 ryx
y
Replace ‘x’ with (x – 2)
222)2( ryx
r
(2, 0)
Let’s move the circle 2 spaces downward.
How do we change the equation to translate the graph
down 2?
x
222 ryx y
Replace ‘y’ with (y + 2)
222 )2( ryx
r(0, -2)
Is the point on the circle?
Prove that a point is on a circle:The circle below is the graph of:
y
4)2()1( 22 yx
r=2
(1, 2)
x
)32 ,2(
4)2()12( 22 y
Plug in x = 2, solve for ‘y’.
4)2()1( 22 y
3)2( 2 y
32y
Yes is on the circle.)32 ,2(
Solve the system:
9)2()3( 22 yx 452 xxy
Prepare to graph: solve for ‘y’
2)3(9)2( xy
22 )3(9)2( xy
2)3(9)2( xy
2)3(9)2( xy
This is two different equations, which are the upper and lower halves of the circle!
2)3(92 xy2)3(92 xy
0.19- 1.12,- ,64.3 ,04.5 x