Pc Intersections Of Lines

IN TE RS ECT IO N OF LIN ES GRAP HIC AN D ALG EG RA IC

MATH

PR OJECT

1. Find the equation of the line that goes through: a. (-7,-10) & (7,8) b. (-6,8) & (6,-2) 9 y = x −1 7 5 y = − x +3 6

Finding Intersections of Lines Objectives: 1. Discover how to find the intersections of lines graphically and algebraically.

Finding Intersections of Lines Objectives: 2. Solve problems by finding the intersections of lines.

Two ways to find intersections of lines. 1. Graph the lines…..

What is the point of intersection of the following lines:

y = 3x − 5 4 x − 3 y = 10

y = 3 x − 5 1. Graph the lines. 4 x − 3 y = 10 y=3x-5

Solve 4x-3y=10 for y. 4x-3y=10 -4x -4x -3y= -4x+10

y = 3 x − 5 1. Graph the lines. 4 x − 3 y = 10 y=3x-5

-3y= -4x+10 -3y= -4x+10 -3 -3

4 10 y = x− 3 3

y = 3 x − 5 1. Graph the lines. 4 x − 3 y = 10 y=3x-5

This appears to be the point of (1,-2) intersection.

Two ways to find intersections of lines. 1. Graph the lines. 2. Use algebra when graphing does not give you an accurate answer.

Find the intersection of the following lines. y=4x-7 12x+2y=1 Since one of the equations is already solved for y, plug that y value into the other equation and solve for “x”.

y=4x-7 12x+2y=1

12x+2(4x-7)=1

12x+8x-14=1 3 x= 4

20x=15

You know the x-value. Plug the x-value into either equation and solve for “y”. y=4x-7 12x+2y=1

3 x= 4

y=4x-7 3 x= 4

3 y=4( )-7 4

y=

12 4

-7

y=3-7 y=-4

The point of intersection is: 3 x= 4

y=-4

3 ( ,−4) 4

or…

You graph these two lines and see if this looks good. y=4x-7 12x+2y=1 The point of 3 intersection ( ,−4) 4 is:

y=4x-7 12x+2y=1

1 y = −6 x + 2 Point of intersection.

3 ( , − 4) 4

Graph the two equations below. Estimate the point of intersection.

y=3x+5 & y=-6x-4 When finished solving graphically, solve algebraically. Compare your two answers. Which is more accurate?

When you solve graphically….

y=3x+5 y=-6x-4 The point of intersection looks like it is at (-1,2). Now check algebraically to see if this is correct.

y=3x+5 y=-6x-4 Since both 3x+5 and –6x-4 are equal to “y”, set them equal to each other and solve for “x”. 3x+5=-6x-4

3x+5=-6x-4 3x + 5 = -6x - 4 +6x - 5 +6x -5 9x = -9 x = -1

Since x = -1, plug –1 in for x and solve for y in either equation. y=3x+5

y=-6x-4 y=-6(-1) – 4 y= 6 – 4 = 2

When we graphed we got (-1, 2). When we solved algebraically, we got (-1,2). This is the correct answer!

The wrestling club is planning a t-shirt sale to raise funds for a tournament in Hawaii. The tshirt company charges an $80.00 set-up fee and $5.00 for each t-shirt. The wrestling club will sell the shirts for $15.00 each.

- t-shirt company charges $80.00 set-up fee and $5.00 for each t-shirt - Club sells each shirt for $15.00.

Write an equation to represent cost and an equation to represent profit and graph each.

COST 80.00 set-up fee+5.00 per shirt y=5.00x+80 PROFIT 15.00 PER SHIRT y=15.00x

COST y=5.00x+80 PROFIT y=15.00x What does “x” stand for? -Number of shirts sold.

COST y=5.00x+80 PROFIT y=15.00x What does “y” stand for? -either profit or cost

COST y=5.00x+80 PROFIT y=15.00x How many shirts must be sold before it starts making a profit?

COST y=5.00x+80 PROFIT y=15.00x Graph each equation, the point of intersection is the break-even point.

S a l e s

BREAK-EVEN POINT

It’s hard to tell the exact coordinates of the break-even $ # of shirts point, so now you will calculate the point-of-intersection algebraically.

COST y=5.00x+80

PROFIT y=15.00x

5x+80=15x 80 = 10x x=8 Now calculate the “y” value by putting 8 in for “x”.

COST y=5.00x+80

PROFIT y=15.00x

y=15.00(8) y=120 The coordinates of the breakeven point are (8,120).

Every shirt sold after 8 is profit.