System of Equations
What is a solution to a system of equations?
A solution to a system of equations is an ordered pair (x, y) that is a solution to BOTH equations in the system.
Find the solution to the following system:
x = y
y = 2x - 4
(4, 4)
What does it mean for a number to be a solution to an equation?
A number is a solution to an equation when you plug that number into the equation it makes the equation true
The knitting club sold 40 scarves and hats at a winter festival and made $700 from the sales. They charged $18 for each scarf and $14 for each hat.
If x represents the number of scarves sold and y represents the number of hats sold, Write a system of equations that represents the constraints in this situation.
x + y = 40
18x + 14y = 700
What do you call two equations that have the same solutions?
equivalent equations
What are three ways to find solutions to systems of equations?
Graphing, substitution, elimination
Find the solution to the following system:
y = 2x + 1
y = 6x + 9
(-2, -3)
Find the solution to the following system:
2x - 4y = 2
x + 4y = 7
(3, 1)
At a restaurant the cost for a breakfast taco and a small glass of milk is $2.10. The cost for 2 tacos and 3 small glasses of milk is $5.15. Which pair of equations can be used to determine t, the cost of a taco, and m, the cost of a small glass of milk?
t + m = 2.10
2t + 3m = 5.15
What does slope-intercept form look like?
y = mx + b
When we graph a linear system (a system with two equations and two variables) the possible cases for solutions are...
no solutions, 1 solution, infinitely many solutions
Find the solution to the following system:
3x + 3y = 15
3x + 2y = 1
(-9, 14)
Find the solution to the following system:
5x + 2y = 10
x + y = 2
(2, 0)
Ryan spends $24 on sodas and cookies. Each soda cost $3.25 and each cookie costs $1.65. The number of sodas Ryan bought is the same as the number of cookies he bought. Write a system to represent this situation. Make sure to define your own variables
x = number of sodas, y = number of cookies
3.25x + 1.65y = 24
x = y
Find the y-intercept of the following equation:
y = -2x + 12
(0, 12)
How can you use a graph to find the solutions to a system of equations?
Graph the two equations and where the solutions are where they intersect
Is (2, 1) a solution to the system
x + y = 3
y = 5x - 9
Yes
Find the solution to the following system:
7x + 3y = 2
2y - x = -10
(2, -4)
An animal shelter takes care of dogs and cats. In total, there are 48 legs. Also, the number of dogs is 5 times the number of cats. If x represents the number of cats and y represents the number of dogs, write a system that represents this scenario.
4x + 4y = 48
y = 5x
Find one solution to the following equation:
y = -x - 3
Answers may vary
Systems that have infinitely many solutions means that the equations are _________________.
equivalent, the same
Is (-5, -3) a solution to the system
x + y = -8
2x - 4y = -2
No
Find the solution to the following system:
3x + 4y = 2
5x + 6y = 10
(14, -10)
The length of a rectangle is equal to one less than double the width. The perimeter of the rectangle is 23 square inches. Let l be the length and w be the width. Write a system describing the situation.
l = 2w - 1
2l + 2w = 23
If you find two unique solutions to a system of equation, what can you say about the solutions?
There are infinitely many solutions to the system