Math-3Lesson 7-4

Solving Systems of Equations Graphically

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).

Systems of Equations

Sometimes there are

more than one solution!

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

xx 41312 77

2x

x

x

y

y

413

12

7

7

4)73(log515 2 x

309.2x

4)73(log5

15

2

xy

y

8)3(57 )2( x

535.0x

8)3(5

7

)2(

xy

y

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

Solve the system:

32 xy

42 xy