Solving Systems
Solve by elimination: 3x + 2y = 14 and 3x - 4y = 2.
(x, y) = (2, 4)
Write a system for a phone plan that costs $40 + $0.10 per minute vs a $60 flat rate.
40 + 0.1x = 60 → x = 200 minutes
What is the elimination method?
Adding or subtracting equations to eliminate one variable
What does it mean if two lines intersect?
One solution
What’s the slope of y = 2x + 5?
2
Solve by substitution: y = 2x + 1 and 4x + y = 11.
(x, y) = (2, 5)
Two days of ticket sales: Write equations for 50a + 30s = 510 and 60a + 20s = 540. Solve for a and s.
a = 7.50, s = 4.50
What is the substitution method?
Replacing one variable with an equivalent expression
What if the lines are parallel?
No Solution
If y = ½x + 4, what’s the y-intercept?
4
What does it mean if you get 0 = 0 after solving a system?
Infinitely many solutions (same line)
Write a system to represent babysitting with $20 flat fee + $12/hr and another sitter who charges $16/hr with no fee.
y = 12x + 20 and y = 16x
Why can you add two equations together?
Because both are true, and adding them keeps the equality true
What if the lines are the same?
Infinitely many solutions
Write a system that crosses at (3, -2)
Write a system that crosses at (3, -2)
What does it mean if you get 0 = 5 after solving a system?
No solution (Parallel lines)
An adult ticket costs $8 and a student ticket costs $5. Write an expression for the total cost if someone buys a adult tickets and s student tickets.
8a + 5s
What does it mean for two equations to share a solution?
They intersect at the same (x, y) point
How do you find a solution from a graph?
Find where the two lines cross
Solve: 2x - y = 10 and 3x + y = 5
(x, y) = (3, -4)
Question: Create your own system that has exactly one solution.
Any two lines that intersect once
(e.g. y = x + 2, y = -x + 4)
A pizza shop sells a 10-inch cheese pizza for $8. Each additional inch adds $1.50, and each topping adds $0.75.
Write an equation that represents the total cost C of a pizza with diameter d inches and t toppings.
C = 8 + 1.5(d−10) + 0.75t
How is elimination connected to graphing?
Both find the intersection point of two lines
Create and graph your own system that has no solution.
Example: y = 2x + 3 and y = 2x - 1 (parallel lines)
Solve: 5x + 4y = 8 and 10x - 4y = 46
x = 3.6
y = -2.5