P.o.D. – Solve each inequality.

1.) 14<2x-6 and 2x-6<50 or {14<2x-6<50}

2.) x-6<-12 or 2x-4>10

3.) 13<y+3(-4y+8)

1.) 14+6<2x<50+6  20<2x<56  10<x<28

2.)

X<-12+6
X<-6
X<-6 or / 2x>10+4
2x>14
x>7

3.)13<y-12y+24  13<-11y+24  -11<-11y 1>y or y<1

5-2: Solving Systems Using Tables, Graphs, or a CAS (Computerized Algebraic Solver)

Learning Target: be able to recognize properties of systems of equations; use systems of two linear equations to solve real-world problems; estimate solutions to systems by graphing.

System:

A set of two or more equations.

Solution Set of a System:

The intersection of the two equations.

Find Solutions Graphically:

  1. Put the equations in slope-intercept form. (Solve for y)
  2. Graph the two equations.
  3. Find the point of intersection.

EX: Solve the system

Step 1: Put each equation in slope-intercept (y=mx+b) form.

Step 2: Graph the two equations.

Step 3: Find the point of intersection.

The solution is (-6,-16).

EX: Fred wants to enclose a 400 square meter rectangular garden with 70 meters of fencing. To do this, he must use one side of his barn as a side of the garden. What can the dimensions of the garden be?

x

Let x+2y=70 and xy=400.

Step 1: Solve both equations for y.

x+2y=70
2y=70-x
Y=(70-x)/2 / xy=400
y=400/x

Step 2: Graph the equations.

Step 3: Find the point(s) of intersection

Because x and y must be positive, we only want to examine Quadrant I. The solutions (x,y) are near (14,28) and (56,7). Thus, the dimensions of the garden should be 56 by 7 or 14 by 28.

EX: Solve the system

Step 1: Solve each equation for y.

xy=30
y=30/x / x-y=1
-y=1-x
y=x-1

Step 2: Graph each equation.

Step 3: Find the point(s) of intersection.

(-5,-6) and (6,5)

We can also find solutions to a system using the TABLE.

-We simply look for where Y1 and Y2 have the same value.

Do the following on your own. Solve.

1.)

2.)

1.)(-0.5, 0.25), (2,4)

2.)(-1.4,-6.9), (10.4, .96)

EX: Have a student come to the board to solve the system

Set the two equations equal to y.

Y= -2x+1 and y=3x-6

(1.4, -1.8)

EX: To make yearbooks, it costs the school $8 per yearbook and a $5000 set-up fee. We sell the yearbooks for $50 each. How many yearbooks must we sell in order to break even?

Set up an equation for cost and another for revenue.

Y=8x+5000 {cost}

Y=50x {revenue}

Graph the two equations. [X axis: 0 to 200; Y axis: 0 to 10,000]

We need to sell 120 yearbooks to begin making a profit.

Upon completion of this lesson, you should be able to:

  1. Set equations equal to y.
  2. Graph multiple equations and find their point(s) of intersection using a graph (calculator).
  3. Model real-world situations using systems of equations.

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HW Pg. 310 4-21