Order of Operations
2 + 3(4 - 1)
11
Solve 4x -7=21
4x=28
x=4
What are the ways to solve a system?
Graphing
Substitution
Elimination
Cramer's Rule
What are the three ways to solve a quadratic?
factoring
completing the square
quadratic formula
Why is graphing helpful?
gives a visual of the algebraic representation
helps find solutions
52-3(2)+(8/4)
25-6+2
19+2
21
What are the 3 forms of linear equations?
Slope Intercept form y=mx+b
Point Slope y-y1=m(x-x1)
Standard Ax+By=C
Solve the system:
2x+y=11
x-y=1
using elimination add the equations
3x=12
x=4
using substitution
4-y=1 so y=3
Find the roots of
x2 - 5x + 6 = 0
(x-6)(x+1)=0
x=6 or x= -1
Solve the system by graphing:
x+y=8
x-y=4
check graphs for solution (6,2)
6 ÷2(1 + 2)
6/2(3)
6/6=1
Put into standard form:
y=3x-10
-3x+1y=-10
Solve the system:
y = 2x - 5
3x + 4y = 14
using substitution
3x + 4(2x-5) = 14
3x +8x-20 = 14
11x=34
x=34/11
y=2(34/11)-5
y=68/11-55/11= 13/11
Solve by completing the square
x2 + 4x - 5 = 0
x2 + 4x - 5 = 0
x2 + 4x +4 =5+4
(x+2)2=9
x+2= +9 or -9
x=7 or -11
Graph y-1= 3/4(x+5)
check graph
-10 ÷ (20 ÷ 2² × 5 ÷ 5) × 8 - 2
-10 ÷ (20 ÷ 2² × 5 ÷ 5) × 8 - 2
-10 ÷ (20 ÷ 4 × 5 ÷ 5) × 8 - 2
-10 ÷ (5 × 5 ÷ 5) × 8 - 2
-10 ÷ (5) × 8 - 2
-2x8-2
-16-2= -18
Solve for the slope & y-intercept
y+2=1/2(x-10)
y=1/2x-7
slope=1/2
y int= -7
A furniture store sells two couches and three tables for $6000. The same store sells five couches and five tables for $12,500. How much does each piece of furniture cost? Let x represent the cost of one couch, and let y represent the cost of one table.
2x + 3y = 6000
5x + 5y = 12500
(2) -> 10x + 10y = 25000
(- 5) -> - 10x - 15y = - 30000
- 5y = - 5000 y = $1000 per table
2x + 3(1000) = 6000
2x + 3000 = 6000
2x = 3000 x = $1500 per couch
Couch = $1500
Table = $1000
Solve the quadratic equation:
3a(a+2) + 1 = 0
3a2+6a +1=0
not factorable so complete the square or QF
x=(-3+√6)/3
Graph the quadratic equation:
(x-1)(x+1)= 5(x+1)
check graphs for zeros & vertex
x=6 & x=-1 vertex x=3/2 y=3/4
10 × 4 - 2 × (4² ÷ 4) ÷ 2 ÷ 1/2 + 9
10 × 4 - 2 × (16 ÷ 4) ÷ 2 ÷ 1/2 + 9
10 × 4 - 2 × (4) ÷ 2 ÷ 1/2 + 9
40-8+9
32 + 9
= 41
Solve (2y+3)/5=4
2y+3= 20
2y=17
y=17/2=8.5
A tennis court has a perimeter of 212 feet. If the length is 24 feet longer than the width, what are the dimensions of the court?
2l + 2w = 212 L = 24 + W
2(24 + W) + 2w = 212
48 + 2w + 2w = 212
4w=164 w= 41 ft
L=24 +41= 65 ft
Solve the quadratic by completing the square
4q2 +12q +9 = 0
Divide everything by 4
q2+3q = -9/4
q2+3q +9/4 = -9/4 +9/4
(q+3/2)2=0
q=-3/2
y>0
3x+1y>6
y>3x
x<7
y<6
check overlapping in 1st quadrant triangluar region
all dotted lines!