Examine the six different equations that follow:
𝑦=5𝑥−4
𝑦=−2𝑥+3
𝑦=3𝑥+1
2𝑦=6𝑥+2
𝑦−5𝑥=2
𝑦−4𝑥+1=0
Complete steps a. through e. to determine which pair of equations form a system with one solution, no solution, or infinite solutions.
Solve each equation and make sure it is in 𝑦=𝑚𝑥+𝑏
y=mx+b
format.
Graph all six equations on the coordinate grid and label them.
An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1.
Circle the two equations that form a system with infinite solutions.
Put a square around the two equations that form a system with no solution.
Underline the two equations that form a system with one solution.
What do you notice about equations that are in each of these categories? Use complete sentences to answer the questions that follow.
Question 2: Infinite Solutions
Consider the two equations you circled, which form a system with infinite solutions. Graph them together on the following coordinate grid.
An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1.
What do you notice about these graphs? What does it mean for a system to have infinite solutions? What do the graphs of a system with infinite solutions look like? Explain what it means graphically for a system to have infinite solutions.
Plug a few points into the original equations and check to show that the system of equations has more than one solution. Explain your work.
Question 3: No Solution
Consider the two equations you put a square around, which form a system with no solution. Graph them together on the following coordinate grid.
An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1.
What do you notice about these equations? What does it mean for a system to have no solution? Explain what it means graphically for a system to have no solution.
Plug a few points into the original equations and check to show that the system of equations has no solution. Explain your work.
Question 4: One Solution
Consider the two equations you underlined, which form a system with one solution. Graph them together on the following coordinate grid.
An illustration of a coordinate plane with four quadrants shows the x and y axes extending from negative 10 to 10 in increments of 1.
What do you notice about these equations? What does it mean for a system to have one solution? Explain what it means graphically for a system to have one solution.
Plug a few points into the original equations and check to show that the system of equations has one solution. Explain your work.
a. All of the equations are already in y = mx + b format, except the last two:
y = 5x - 4
y = -2x + 3
y = 3x + 1
y = (6x + 2)/2 = 3x + 1
y = 5x - 2
y = 4x - 1
b. Graph all six equations on the coordinate grid and label them:
[Here, the AI cannot produce a graph, so you should graph these equations manually on your coordinate grid or use a graphing tool like Desmos.]
c. When examining the graph, we can see that:
- Equations 3 (y = 3x + 1) and 4 (y = 3x + 1) have infinite solutions (circle these equations).
- Equations 1 (y = 5x - 4) and 2 (y = -2x + 3) have one solution (underline these equations).
- Equations 5 (y = 5x - 2) and 6 (y = 4x - 1) have no solution (put a square around these equations).
d. What we notice about the equations in each category:
- Infinite solutions: The equations have the same slope and y-intercept; they are the same line on the graph.
- One solution: The equations intersect at one point; different slopes and y-intercepts.
- No solution: The equations are parallel, meaning they have the same slope but different y-intercepts.
Question 2: Infinite Solutions
Graph equations 3 (y = 3x + 1) and 4 (y = 3x + 1) together:
[Graph these equations manually or use Desmos.]
We notice that the graphs of these equations overlap, as they are the same line. For a system to have infinite solutions, it means that the two equations represent the same line. The graphs of a system with infinite solutions look identical and overlap completely.
For example, plugging in x=1, we get y = 3 + 1 = 4 for both equations. Similarly, for x=2, we get y = 6 + 1 = 7 for both equations. This shows that the system has more than just one solution.
Question 3: No Solution
Graph equations 5 (y = 5x - 2) and 6 (y = 4x - 1) together:
[Graph these equations manually or use Desmos.]
We notice that the graphs of these equations are parallel, meaning that they never intersect. For a system to have no solution, it means that the lines never intersect or are parallel. Graphically, this means that they have the same slope but different y-intercepts.
For example, plugging in x=1 for both equations, we get y = 5(1) - 2 = 3 for equation 5 and y = 4(1) - 1 = 3 for equation 6. For x=2, we get y = 5(2) - 2 = 8 for equation 5 and y = 4(2) - 1 = 7 for equation 6. These results are different, showing that the system has no solution.
Question 4: One Solution
Graph equations 1 (y = 5x - 4) and 2 (y = -2x + 3) together:
[Graph these equations manually or use Desmos.]
We notice that the graphs of these equations intersect at one point. For a system to have one solution, it means that the lines intersect at exactly one point. Graphically, this means that they have different slopes and y-intercepts.
For example, plugging in x=1, we get y = 5(1) - 4 = 1 for equation 1 and y = -2(1) + 3 = 1 for equation 2. This shows that the system has one solution at the point (1, 1).