Solving Systems by Graphing

Show your work for each problem.

1.
What does the solution of a system of equations represent on a graph?
1. a)The point where one line crosses the x-axis
2. b)The y-intercept of the first line
3. c)The slope of the lines
4. d)The point where both lines intersect
Answer: The point where both lines intersect
The solution of a system is the point where both lines intersect. This point satisfies both equations simultaneously.
2.
If two lines are parallel, how many solutions does the system have?
1. a)No solution
2. b)Infinitely many solutions
3. c)One solution
4. d)Two solutions
Answer: No solution
Parallel lines have the same slope but different y-intercepts. Since they never intersect, there is no point that satisfies both equations, so the system has no solution.
3.
The lines and intersect at point . What is the y-coordinate?
Answer: 4
Substituting into gives . You can verify: also gives 4.
4.
Find the x-coordinate where and intersect.
Answer: 3
Setting and solving: , so . The intersection point is .
5.
Which system has infinitely many solutions?
1. a) and
2. b) and
3. c) and
4. d) and
Answer: and
Simplifying by dividing by 2 gives , which is the same as the first equation. They are the same line!
6.
Solve the system by graphing:
Answer: (2, 4)
1. Set the equations equal to find : . Solve for . 2
2. Substitute into the first equation to find . 4
3. Write the solution as an ordered pair . (2, 4)
7.
The system and has how many solutions? (Enter 0, 1, or infinity)
Answer: 0
Both lines have slope 3 but different y-intercepts (-2 and 5). They are parallel and never intersect, so there are 0 solutions.
8.
Solve the system:
Answer: (2, 3)
1. Set . Add to both sides. What do you get? 3x - 1 = 5
2. Add 1 to both sides. What is equal to? 6
3. Divide by 3. What is ? 2
4. Substitute into . What is ? 3
9.
A company's costs are and revenue is . At what quantity do they break even?
1. a)50
2. b)25
3. c)10
4. d)20
Answer: 20
Setting : Subtract to get , divide by 5 to get . They break even at 20 units.
10.
Car A starts at position 20 km and travels at 60 km/h (). Car B starts at position 0 and travels at 80 km/h (). When do they meet?
Answer: 1 hour at 80 km
1. Set the positions equal: . Subtract from both sides. 20 = 20x
2. Solve for (time in hours). 1
3. Find the position using when . 80
11.
Find the x-coordinate of the solution: and
Answer: 4
Setting : Subtract to get , add 3 to get , multiply by to get .
12.
Supply: . Demand: . Find the equilibrium price (y-value).
Answer: 15
1. Set supply equal to demand: . Add to both sides. 3x + 5 = 20
2. Subtract 5 from both sides. 3x = 15
3. Solve for (equilibrium quantity). 5
4. Substitute into to find the equilibrium price. 15
13.
Which statement is TRUE about the system and ?
1. a)The system has exactly one solution
2. b)The system has no solution
3. c)The system has infinitely many solutions
4. d)The system has exactly two solutions
Answer: The system has infinitely many solutions
Both equations are identical: . They represent the same line, so every point on that line is a solution. This means infinitely many solutions!
14.
Two lines with slopes and but different y-intercepts will have how many solutions?
Answer: 0
Lines with the same slope (both = 3) but different y-intercepts are parallel. Parallel lines never intersect, so there are 0 solutions.
15.
Verify that is the solution of the system:
Answer: Yes, verified
1. Substitute into . What is ? 5
2. Does this match the y-coordinate? (yes/no) yes
3. Substitute into . What is ? 5
4. Does this also match? Is the solution? (yes/no) yes