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 $(x, y)$ 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 $y = x + 3$ and $y = -x + 5$ intersect at point $(1, ?)$ . What is the y-coordinate?
Answer: 4
Substituting $x = 1$ into $y = x + 3$ gives $y = 1 + 3 = 4$ . You can verify: $y = -x + 5 = -1 + 5 = 4$ also gives 4.
4.
Find the x-coordinate where $y = 2x$ and $y = x + 3$ intersect.
Answer: 3
Setting $2x = x + 3$ and solving: $2x - x = 3$ , so $x = 3$ . The intersection point is $(3, 6)$ .
5.
Which system has infinitely many solutions?
1. a) $y = 2x + 1$ and $y = -x + 4$
2. b) $y = 2x + 1$ and $y = 2x + 3$
3. c) $y = 2x + 1$ and $2y = 4x + 2$
4. d) $y = 2x + 1$ and $y = 3x + 1$
Answer: $y = 2x + 1$ and $2y = 4x + 2$
Simplifying $2y = 4x + 2$ by dividing by 2 gives $y = 2x + 1$ , which is the same as the first equation. They are the same line!
6.
Solve the system by graphing. Plot the points of each line for $x = 0, 1, 2$ :
$\begin{cases} y = x + 2 \\ y = -x + 6 \end{cases}$
Answer: (2, 4)
1. Line 1, $y = x + 2$ , passes through $(0, 2)$ and $(1, 3)$ . What is $y$ at $x = 2$ ? 4
2. Line 2, $y = -x + 6$ , passes through $(0, 6)$ and $(1, 5)$ . What is $y$ at $x = 2$ ? 4
3. Both lines pass through the same point. Write the intersection point as $(x, y)$ . (2, 4)
7.
How many solutions does the system $y = 3x - 2$ and $y = 3x + 5$ have? Type the number.
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:
$\begin{cases} y = 2x - 1 \\ y = -x + 5 \end{cases}$
Answer: (2, 3)
1. Set $2x - 1 = -x + 5$ . Add $x$ to both sides. What do you get? 3x - 1 = 5
2. Add 1 to both sides. What is $3x$ equal to? 6
3. Divide by 3. What is $x$ ? 2
4. Substitute $x = 2$ into $y = 2x - 1$ . What is $y$ ? 3
9.
A company's costs are $y = 100 + 5x$ and revenue is $y = 10x$ . At what quantity $x$ do they break even?
1. a)50
2. b)25
3. c)10
4. d)20
Answer: 20
Setting $100 + 5x = 10x$ : Subtract $5x$ to get $100 = 5x$ , divide by 5 to get $x = 20$ . They break even at 20 units.
10.
Car A starts at position 20 km and travels at 60 km/h ( $y = 60x + 20$ ). Car B starts at position 0 and travels at 80 km/h ( $y = 80x$ ). When do they meet?
Answer: After 1 hour, at the 80 km mark
1. Set the positions equal: $60x + 20 = 80x$ . Subtract $60x$ from both sides. 20 = 20x
2. Solve for $x$ (time in hours). 1
3. Find the position using $y = 80x$ when $x = 1$ . 80
11.
Find the x-coordinate of the solution: $y = \frac{1}{2}x + 3$ and $y = 2x - 3$
Answer: 4
Setting $\frac{1}{2}x + 3 = 2x - 3$ : Subtract $\frac{1}{2}x$ to get $3 = \frac{3}{2}x - 3$ , add 3 to get $6 = \frac{3}{2}x$ , multiply by $\frac{2}{3}$ to get $x = 4$ .
12.
Supply: $y = 2x + 5$ . Demand: $y = -x + 20$ . Find the equilibrium price (y-value).
Answer: 15
1. Set supply equal to demand: $2x + 5 = -x + 20$ . Add $x$ to both sides. 3x + 5 = 20
2. Subtract 5 from both sides. 3x = 15
3. Solve for $x$ (equilibrium quantity). 5
4. Substitute $x = 5$ into $y = 2x + 5$ to find the equilibrium price. 15
13.
Which statement is TRUE about the system $y = -2x + 7$ and $y = -2x + 7$ ?
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: $y = -2x + 7$ . They represent the same line, so every point on that line is a solution. This means infinitely many solutions!
14.
The lines $y = 3x + 1$ and $y = -2x + 4$ have different slopes. How many solutions does the system have?
Answer: 1
The slopes $3$ and $-2$ are different, so the lines are not parallel and not the same line. They cross at exactly one point, so the system has 1 solution.
15.
Verify that $(3, 5)$ is the solution of the system:
$\begin{cases} y = 2x - 1 \\ y = -x + 8 \end{cases}$
Answer: Yes, $(3, 5)$ is the solution
1. Substitute $x = 3$ into $y = 2x - 1$ . What is $y$ ? 5
2. Does this match the y-coordinate? yes
3. Substitute $x = 3$ into $y = -x + 8$ . What is $y$ ? 5
4. Does this also match? Is $(3, 5)$ the solution? yes