Concepts
Standard Form.
What is 0/10?
0.
Which variable should we eliminate?
3x - y = 5
2x + y = 4
y, because it has opposite coefficients.
Set up the systems of equations:
If two numbers have a sum of 10, and a difference of 2, what are my two numbers?
x + y = 10
x - y = 2
What is slope-intercept form?
y = mx + b
(x - 2) / -1
Which type of coefficients (equal or opposite) can you see here?
2x + 4y = 3
3x + 4y = 12
The y variables have equal coefficients.
Set up the systems of equations:
Mark sold overall 120 cups of two different flavors of lemonade today. Regular lemonade cost $3. Raspberry lemonade cost $4. He made $440 today. How many cups of each flavor did he sell?
x + y = 120
3x + 4y = 440
What is standard form?
Ax + By = C
Solve for x:
10x = 0
x = 0
After we find the first variable, how do we find the second variable?
Plug the variable you found back in to an original equation.
Set up a systems of equations:
Flying to HK with a tailwind, a plane averaged 168 km/h. On the return, flying into the same wind, the plane averaged 110 km/h. What is the speed of the wind and the speed of the plane in still air?
x + y = 168
x - y = 110
How do we eliminate variables with equal coefficients?
We subtract the two equations.
Add:
x + 3y =3
(+) x - 4y = 4
2x - y = 7
What should we do if there are no equal or opposite variables?
Example:
4x + 3y = 10
3x + 5y = 12
Pick one variable (x or y) and then find the lowest common multiple. Multiply both equations by the number required to get to the LCM for that variable.
Example:
(4x + 3y = 10) x3 = (12x + 9y = 30)
(3x + 5y = 12) x4 = (12x + 20y = 48)
Set up the systems of equations:
Matt sold 3 small boxes of oranges and 14 large boxes of oranges for a total of $203. Ming sold 11 small boxes of oranges and 11 large boxes of oranges for a total of $220. Find the cost of one small box of oranges and one large box of oranges.
3x + 14y = 203
11x + 11y = 220
How do we eliminate variables when we have opposite coefficients?
We add the two equations.
Subtract:
x - 2y = 5
(-) -x - 4y = 6
2x + 2y = -1
Solve for x and y:
3x + 2y = 8
2x - 4y = 16
(4, -2)
Set up the systems of equations:
On a trip to NYC, High School A rented 1 van and 6 buses to carry 372 students. High School B rented 4 vans and 12 buses to carry 780 students. Each van carries the same number of students. Each bus carries the same number of students. How many students fit into a bus? Into a van?
x + 6y = 372
4x + 12y = 780