Graphing
The y-intercept and resulting coordinates of:
y = 3x + 6
y -intercept = 6
coordinates = (0,6)
Find the point of intersection:
y = 2x + 4
y = 4x + 6
2x + 4 = 4x + 6 y = 2(-1) + 4
2x - 4x = 6 - 4 y = -2 + 4
-2x = 2 y = 2
x = 2/-2 POI (-1,2)
x = -1
Solve by method of elimination:
3x + y = 19
4x - y = 2
Step 1:
3x + y =19
+ 4x - y = 2
7x = 21
x= 21/7
x = 3
Step 2:
3(3) + y = 19
y = 19 - 9
y = 10
POI (3,10)
Are these two equations equivalent - yes or no?
12x + 6y = 9
4x + 2y = 3
YES.
Divide by 3 to get from equation 1 to equation 2.
What the slope determines on a line.
The slope of:
y = 3/4x + 8
rise / run = 3/4
Find the point of intersection:
y = 9x + 3
y = 6x + 12
9x - 6x = 12 - 3 y = 27 + 3
3x = 9 y = 30
x = 9/3 POI = (3,30)
x = 3
Solve by method of elimination:
2x + 4y = 24
6x - 4y = 8
Step 1:
2x + 4y = 24
+ 6x - 4y = 8
8x = 32
x = 32/8
x = 4
Step 2:
2(4) + 4y = 24
8 + 4y = 24
4y = 24 - 8
4y = 16
y = 16/4
y = 4
POI (4,4)
Are these two equations equivalent - yes or no?
20x + 10y = 5
5x + 2y = 1
NO.
The y-intercept and slope of:
2y = 6x + 4
y-intercept = 2
slope = 3/1
Find the point of intersection:
y = 15x + 30
y = 45x + 90
15x + 30 = 45x + 90 y = 15(-2) + 30
15x - 45x = 90 - 30 y = -30 + 30
-30x = 60 y = 0
x= 60 / -30 POI (-2,0)
x= -2
Solve by method of elimination:
x + 3y = 7
x + y = 3
Step 1:
x + 3y = 7
x + y = 3 (multiply by 3 to get 3y)
Step 2:
x + 3y = 7
- 3x +3y = 9
-2x = 2
x = 2/-2
x = -1
Step 3:
(-1) + 3y = 7
3y = 7 + 1
3y = 8
y = 8/3
POI (-1, 8/3)
Find an equivalent equation to:
5x + 3y = 10
10x + 6y = 20
The y-intercept and slope of:
3y = 9x + 2
y-intercept = 2/3
slope = 3/1
Find the point of intersection:
3y = 3x + 9
4y = -4x + 8
Step 1:
y = 3/3x + 9/3
y = 1x + 3
y = -4/4x + 8/4
y = -1x + 2
Step 2:
1x + 3 = -1x + 2
1x + 1x = 2 - 3
2x = -1
x = -1/2
Step 3:
y = 1x + 3
y = 1(-1/2) + 3
y = -0.5 + 3
y = 2.5
POI (-1/2, 2.5)
Solve by method of elimination:
4x + 3y = 13
5x - 4y = -7
Step 1:
4x + 3y = 13 (multiply by 5 to make 20x)
5x - 4y = -7 (multiply by 4 to make 20x)
20x + 15y = 65
- 20x - 16y = -28
31y = 93
y = 93/31
y = 3
Step 2:
4x + 3(3) = 13
4x + 9 = 13
4x = 13 - 9
4x = 4
x = 4/4
x = 1
POI(1, 3)
Find two equivalent equations to:
2x + 6y = 12
Multiply by 2:
4x + 12y = 24
Divide by 2:
1x + 3y = 4
The y-intercept and slope of:
5y = 7x + 40
slope = 7/5
Find the point of intersection:
10y = 2x + 4
5y = 4x + 6
y = 2/10x + 4/10
y = 1/5x + 2/5
y = 4/5x + 6/5
Step 2:
1/5x + 2/5 = 4/5x + 6/5
1/5x - 4/5x = 6/5 - 2/5
-3/5x = 4/5
x = (-3/5) / (4/5)
x = (-3/5) x (5/4)
x= -15/20
x = -3/4
Step 3:
y = 1/5(-3/4) + 2/5
y = -3/20 + 2/5
y = -3/20 + 8/20
y = 5/20
y = 1/4
POI (-3/4, 1/4)
Solve the linear system by method of elimination:
4x + 6y = 8
10x - 6y = 8
4x + 6y = 8
+ 10x - 6y = 8
14x = 16
x= 16/14
x = 8/7
Step 2:
4(8/7) + 6y = 8
32/7 + 6y = 8
32/7 - 8 = -6y
32/7 - 56/7 = -6y
-24/7 = -6y
(-24/7) / -6 = y
(-24/7) x (-1/6) = y
-24/42 = y
-12/21 = y
-4/7 = y
Find three equivalent equations to:
7x + 21y = 56
Divide by 7:
x + 3y = 8
Multiply by 2:
14x + 42y = 112
Multiply by 3:
21x + 63y = 168