Graphing Systems
What is the solution to a system of equations represented by the point where the two lines intersect on a graph?
The point of intersection.
What is the first step when using substitution to solve y = 2x + 1 and 3x + y = 7?
Substitute 2x + 1 for y in the second equation.
What is the goal of elimination?
To eliminate one variable by adding or subtracting the equations.
What does it mean if elimination results in a false equation like 0 = 5?
The system has no solution (parallel lines).
Two movie tickets and one soda cost $22. One movie ticket and one soda cost $12. How much does one movie ticket cost?
$10
True or False: If two lines on a graph are parallel, they have one solution.
False (they have no solution).
Solve using substitution: y = x + 2 and 3x + 2y = 10.
(2,4)
Solve using elimination: 2x + y = 7 and 4x - y = 5.
(3, 1)
What does it mean if elimination results in a true equation like 0 = 0?
The system has infinitely many solutions (same line).
The sum of two numbers is 20. Their difference is 4. What are the two numbers?
12 and 8
Graph the system: y = 2x + 3 and y = (-x) + 1. What is the solution?
(-2, -1)
Solve using substitution: x = 2y - 3 and 3x - y = 7.
x= 17/5, y = 16/5
Solve using elimination: 5x + 2y = 16 and 3x - 2y = 4.
(5/2, 7/4)
Solve using elimination: 3x - 6y = 12 and x - 2y = 4. What is the solution?
Infinite solutions (same equation).
A school sells adult and student tickets for a play. Adult tickets are $5 and student tickets are $3. If 50 tickets were sold for $200, how many student tickets were sold?
25
How can you estimate the solution of a system by graphing?
By identifying the point where the two lines cross.
When is substitution the best method to use?
When one equation is already solved for a variable (or easily can be).
What must be done before using elimination if the coefficients of a variable don’t match?
Multiply one or both equations to create a common coefficient.
Solve using elimination: 2x + 3y = 6 and 4x + 6y = 12. What is the solution?
Infinite solutions (same equation).
A boat travels 30 miles downstream in 2 hours and 20 miles upstream in 2 hours. What is the speed of the boat in still water?
12.5 mph
If a system has an infinite number of solutions, what must be true about the two equations?
They represent the same line (they are dependent).
Solve the system: y = -2x + 5 and 3x + 2y = 4 using substitution.
(2,1)
Solve using elimination: 2x + 3y = 12 and 4x - 5y = -6.
(3, 2)
Solve using elimination: 5x - 2y = 10 and 10x - 4y = 25. What is the solution?
No solution (parallel lines).
A store sells two different types of gift baskets: Small baskets cost $12, and large baskets cost $18. If the store sold 30 baskets for a total of $450, how many of each type were sold?
15 small, 15 large