Solve each equation and make sure it is in y=mx+by=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.
1. y=5x−4
2. y=-2x+3
3. y=3x+1
4. 2y=6x+2
5. y-5x=2
6. y-4x+1=0
7. y=5x
Equations in y=mx+b format:
1. y = 5x - 4
2. y = -2x + 3
3. y = 3x + 1
4. y = 3x + 1 (Divide by 2)
5. y = 5x + 2
6. y = 4x - 1
7. y = 5x
Graph of the equations on a coordinate grid:
(Insert graph with all equations plotted)
Circle the two equations that form a system with infinite solutions:
- Equations 3 and 4
Square the two equations that form a system with no solution:
- Equations 1 and 6
Underline the two equations that form a system with one solution:
- Equations 2 and 7.
Consider the two equations you circled, which form a system with infinite solutions. Graph them together on the following coordinate grid. 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.
The two equations that form a system with infinite solutions are:
3. y = 3x + 1
4. y = 3x + 1
Graph of equations 3 and 4 on a coordinate grid:
(Insert graph with equations 3 and 4 plotted)
What I notice about these graphs is that they lie directly on top of each other. This means that they completely coincide and intersect at every point. For a system of equations to have infinite solutions, it means that the two equations are equivalent and represent the same line. In other words, any point on one line will also be a solution for the other line.
Graphically, the graphs of a system with infinite solutions overlap perfectly, showing that the lines are the same and have an infinite number of common solutions.
Let's plug in some points to show that the system of equations has more than one solution:
For equation 3: y = 3x + 1
- When x = 1, y = 3(1) + 1 = 4
- When x = 2, y = 3(2) + 1 = 7
For equation 4: y = 3x + 1
- When x = 1, y = 3(1) + 1 = 4
- When x = 2, y = 3(2) + 1 = 7
As we can see, for both equations, when x = 1, y = 4, and when x = 2, y = 7. This confirms that there are multiple solutions, and the two equations have infinite solutions because they represent the same line.
Consider the two equations you put a square around, which form a system with no solution. Graph them together on the following coordinate grid.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.
The two equations that form a system with no solution are:
1. y = 5x - 4
6. y = 4x - 1
Graph of equations 1 and 6 on a coordinate grid:
(Insert graph with equations 1 and 6 plotted)
What I notice about these equations is that they are parallel lines. Parallel lines have the same slope but different y-intercepts, which means they will never intersect, leading to no common solution.
For a system of equations to have no solution, it means that the lines represented by the equations are distinct and will not meet at any point. Graphically, the lines are parallel and do not intersect, indicating that there is no common solution.
Let's plug in some points to show that the system of equations has no solution:
For equation 1: y = 5x - 4
- When x = 1, y = 5(1) - 4 = 1
- When x = 2, y = 5(2) - 4 = 6
For equation 6: y = 4x - 1
- When x = 1, y = 4(1) - 1 = 3
- When x = 2, y = 4(2) - 1 = 7
As we can see, for x = 1, the y-values are different for both equations (1 and 3), and for x = 2, the y-values are different again. This confirms that there is no point where both equations are true, hence no solution.
Consider the two equations you underlined, which form a system with one solution. Graph them together on the following coordinate grid. 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.
The two equations that form a system with one solution are:
2. y = -2x + 3
7. y = 5x
Graph of equations 2 and 7 on a coordinate grid:
(Insert graph with equations 2 and 7 plotted)
What I notice about these equations is that they intersect at a single point, which is the solution to the system. For a system of equations to have one solution, it means that the two lines represented by the equations intersect at exactly one point, providing a unique solution.
Graphically, a system with one solution has two lines that intersect at a single point, indicating the unique solution shared by both equations.
Let's plug in some points to show that the system of equations has one solution:
For equation 2: y = -2x + 3
- When x = -1, y = -2(-1) + 3 = 5
- When x = 0, y = -2(0) + 3 = 3
For equation 7: y = 5x
- When x = -1, y = 5(-1) = -5
- When x = 0, y = 5(0) = 0
As we can see, for x = -1, the y-values are different for both equations (-5 and 5), and for x = 0, the y-values are different again (0 and 3). This confirms that there is only one point where both equations are true, which is the unique solution to the system.
Chose an equation from the list that represents a proportional relationship and another that represents a non-proportional relationship and write them here. graph themWhat 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.
Equation representing a proportional relationship:
7. y = 5x
Equation representing a non-proportional relationship:
1. y = 5x - 4
Graph of equations 7 and 1 on a coordinate grid:
(Insert graph with equations 7 and 1 plotted)
Notice about the similarities of the two graphs:
Both graphs pass through the origin (0,0) and have a positive slope.
Difference between the two graphs:
The graph of the proportional relationship (equation 7) is a straight line that goes through the origin, while the non-proportional relationship (equation 1) is also a straight line but does not pass through the origin.
Graphically, a proportional relationship means that the graph is a straight line passing through the origin. This indicates that the variables are directly proportional and increase or decrease at a constant rate. The slope of the line represents the constant of proportionality.
Non-proportional relationship graphically means that the graph is a straight line but does not pass through the origin. This suggests that the variables in the relationship are not directly proportional to each other, and the increase or decrease is not constant.
Difference between proportional and non-proportional:
A proportional relationship is characterized by a graph that is a straight line passing through the origin and has a constant rate of change, while a non-proportional relationship does not pass through the origin and does not have a constant rate of change.
Real-world scenario for the proportional graph (equation 7):
A real-world scenario that represents the proportional relationship y = 5x could be the cost of renting a car. Let's say a car rental company charges $5 per hour for renting a car. The total cost, y, depends on the number of hours, x, the car is rented. This forms a proportional relationship since the total cost increases by $5 for each additional hour of rental.