Graphing Linear Equations Portfolio
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Investigate and explain the different types of graphical solutions.
Question 1
Examine the different equations that follow:
y=5x−4
y=−2x+3
y=3x+1
2y=6x+2
y−5x=2
y−4x+1=0
y=5x
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
format.
Graph all equations on a coordinate grid (shown below) and label them.
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.
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 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.
Question 5: Proportional and Non-Proportional Relationships
Chose an equation from the list that represents a proportional relationship and another that represents a non-proportional relationship and write them here.
Proportional __________________________
Non-Proportional _______________________
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 the similarities of the two graphs? What do you notice about the difference between the two graphs?
Explain what it means graphically to be proportional. Explain what it means graphically to be non-proportional.
Explain the difference between proportional and non-proportional.
Using your proportional graph, create a real-world scenario that would describe the graph.
Question 1:
The pair of equations that form a system with infinite solutions are:
1) y=5x
2) y=5x
The pair of equations that form a system with no solution are:
1) y=5x
2) y-4x+1=0
The pair of equations that form a system with one solution are:
1) y=5x
2) y=3x+1
For equations that form a system with infinite solutions, they are essentially the same line or have the same slope and y-intercept, resulting in all points on one graph also being on the other graph. For equations that form a system with no solution, the lines do not intersect at any point on the graph. And for equations that form a system with one solution, the lines intersect at a single point on the graph.
Question 2:
For a system to have infinite solutions, the graphs of the equations are essentially the same line, overlapping each other entirely. This means that any point that satisfies one equation will also satisfy the other. Therefore, all points on the graph are solutions for the system.
Plugging in points into the equations will show that each point lies on both lines, confirming that the system has infinite solutions.
Question 3:
For a system to have no solution, the graphs of the equations are parallel lines that never intersect. This means that there is no point that satisfies both equations simultaneously.
Plugging points into the equations will show that there is no common solution between the two lines, confirming that the system has no solution.
Question 4:
For a system to have one solution, the graphs of the equations intersect at a single point. This means that there is only one point where both equations are true simultaneously.
Plugging points into the equations will show that there is only one point that satisfies both equations, confirming that the system has one solution.
Question 5:
Proportional: y=5x
Non-Proportional: y-4x+1=0
The graphs of the proportional and non-proportional equations will both pass through the origin (0,0) since they both have a y-intercept of 0. However, the proportional graph will be a straight line passing through the origin, while the non-proportional graph will be a line with a different slope and y-intercept.
Graphically, proportional relationships show a straight line passing through the origin, indicating a constant rate of change. Non-proportional relationships show a line that is not passing through the origin, indicating a changing rate of change.
The difference between proportional and non-proportional is that proportional relationships have a constant rate of change, resulting in a straight line passing through the origin, while non-proportional relationships have a changing rate of change, resulting in a curved line.
A real-world scenario for the proportional graph could be the relationship between the number of hours worked and the amount of money earned at a fixed hourly rate.