Constructing Systems of Equations
Substitution
Elimination
Linear Inequalities
Vocabulary
100
At Beans & Bagels, a bagel costs $1 more than a cup of coffee. If 4 cups of coffee and 6 bagels cost $31, find the cost of one bagel.
One bagel costs $3.50
100
y = 2
3x + 2y = 10
(2, 2)
100
x + 3y = 5
2x − 3y = −8
(−1, 2)
100
What does a dotted line mean when graphing linear inequalities?
> or <
100
Consists of two or more linear equations in the same variables.
System of linear equations (linear system)
200
A group of 41 people are going to a concert together in 7 vehicles. Some of the vehicles can hold 7 people each, and the rest can hold 5 people each. Assuming all the vehicles are filled to capacity, exactly how many of the vehicles can hold 7 people?
Exactly 3 vehicles can hold 7 people.
200
x = −3y + 2
−x + 2y = 3
**CALC ACTIVE
(−1, 1)
200
−5y + 8x = −18
5y + 2x = 58
(4, 10)
200
How is graphing systems of inequalities different than graphing systems of equations? (2 things)
Sometimes they use dotted lines and you have to shade for inequalities.
200
The method of solving equations where you add or subtract equations to end up with one variable.
Elimination
300
Mike and Sally are math teachers at the same high school. If Mike has 6 more years of teaching experience than Sally has, and the two of them have 16 years of teaching experience combined, how many years of teaching experience does Sally have?
Sally has 5 years of teaching experience.
300
x − 2y = −10
3x = y
(2, 6)
300
3y − 5x = −26
−2y − 5x = −16
**CALC ACTIVE
(4, −2)
300
How do you know where to shade when solving systems of inequalities?
Test a point in the inequality. If it's true, shade toward the point. If false, shade away from the point. The solution is where the shaded regions overlap.
300
How can you tell if a linear system has infinitely many solutions?
The equations are the same, the graphed lines lie on top of one another.
400
You are running a concession stand at a basketball game. You are selling hot dogs and sodas. Each hot dog costs $1.50 and each soda costs $0.50. At the end of the night you made a total of $78.50. You sold a total of 87 hot dogs and sodas combined. How many hotdogs did you sell?
You sold 35 hotdogs.
400
2x – 3y = –2
y = −4x + 24
**CALC ACTIVE
(5, 4)
400
3x − 4y = 8
18x − 5y = 10
(0, -2)
400
What type of line should be used when graphing an inequality with an "or equal to" symbol?
A Solid Line
400
When will a system of linear inequalities have no solution?
The inequalities are parallel meaning they have the same slope but different y-intercepts and shaded away from each other. (Not in opposite directions)
500
You and a friend go to Tacos Galore for lunch. You order three soft tacos and three burritos and your total bill is $11.25. Your friend's bill is $10.00 for four soft tacos and two burritos. How much do soft tacos cost? How much do burritos cost?
Soft tacos cost $1.25 and burritos cost $2.50
500
5x − 13 = y
−5x + y = 24
no solution
500
4x−9y = 2
12x−5y = −38
**CALC ACTIVE
(-4, -2)
500
Based on the graph below, where would we find a solution to the system of inequalities?
Any point that lands in the shaded region
500
How can you tell that a point is a solution to a system of inequalities?
The point must land in BOTH shaded areas (The "common ground")