The Structure of Systems of Equations
How many solutions will this system of equations have?
10x + 2 = y
10x + 4 = y
No solutions
What is the solution to this system of equations?
x = 5
y = x + 2
(5, 7)
What is the solution to this system of equations?
y = 10x
x = 3.5
(3.5, 35)
What would the graph of this system of equations look like?
y = 2x + 5
y = 2x + 0
They would be parallel lines with no point of intersection. They lines would have the same slope, and one line would pass through the origin while the other would pass through the point (0,5)
How many solutions will this system of equations have?
3x + 5 = y
3x + 12 = y
One Solution
What is the solution to this system of equations?
x = 8
y = -11
(8, -11)
What is the solution to this system of equations?
y = 1/2x + 15
x = 15
(15, 22.5)
If you graphed the following system of equations, how many points of intersection would it have?
y + x = 10
y + x = 20
None. This system has no solutions
How many solutions will this system of equations have?
2x + 8 = y
12x + 48 = 6y
Infinite solutions
What is the solution to this system of equations?
y = 3x - 2
y = 4
(2, 4)
What is the solution to this system of equations?
y = 11x + 4
y = 10x + 4
(0,4)
What would the graph of this system of equations look like?
x = 8
y = -11
The graph would intersect at (8, -11). The graph would have a vertical line at going through 8 on the x axis and a horizontal line going through -11 on the y axis
How many solutions will this system of equations have?
x = 5
y = 11
One solution
What is the solution to this system of equations?
y = 2x + 3
y = 1/2 (4x + 3)
No solutions
What is the solution to this system of equations?
y = 5x + 7
y = 4x + 9
(2, 17)
What would the graph of this system of equations look like?
2x + 4 = y
1x + 2 = y
The lines would have one intersection point and both have positive sloped lines.
Mr. Turner says that he can tell just by looking at the structure of a systems of equations how many solutions it will have. What features is he looking for?
One solution: different slopes and vertical intercepts OR different slopes and same vertical intercept.
NO solutions: same slopes, different vertical intercepts
Infinite solutions: same slopes and same vertical intercepts
What is the solution to this system of equations?
y = 1/3 (9x + 3)
y = 3x + 1
Infinite solutions
What is the solution to this system of equations?
4x + 9 = y
5x + 4 = y
(5, 29)
What would the graph of this system of equations look like?
10x + 5 = y
50y = 500x + 250
The lines would be directly on top of one another and have infinite places where they touch