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
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.
Graphing Linear Equations Portfolio Worksheet
Use the rubric to check your work on the portfolio.
To determine which pair of equations form a system with one solution, no solution, or infinite solutions, we need to solve each equation and put it in y=mx+b format:
1. y = 5x - 4
2. y = -2x + 3
3. y = 3x + 1
4. y = 3x + 1 (divide by 2 to get y = 3x + 1/2)
5. y = 5x + 2
6. y = 4x - 1
The equations in y=mx+b format are:
1. y = 5x - 4
2. y = -2x + 3
3. y = 3x + 1
4. y = 3x + 1/2
5. y = 5x + 2
6. y = 4x - 1
Circle the two equations that form a system with infinite solutions:
3. y = 3x + 1
4. y = 3x + 1/2
Square around the two equations that form a system with no solution:
1. y = 5x - 4
5. y = 5x + 2
Underline the two equations that form a system with one solution:
2. y = -2x + 3
6. y = 4x - 1
For question 2:
The two equations form a system with infinite solutions have the same slope and different y-intercepts. This means they are parallel lines and will never intersect. When a system has infinite solutions, it means the two equations represent the same line. The graphs of a system with infinite solutions overlap perfectly.
For question 3:
The two equations form a system with no solution are parallel lines with different slopes. This means they will never intersect. When a system has no solution, it means the two lines never cross each other. Graphically, the lines are parallel and will not intersect at any point.
For question 4:
The two equations form a system with one solution intersect at a single point. This means the two lines intersect only once. When a system has one solution, it means the two lines cross at one point. Graphically, the lines intersect at a single point.
By analyzing the different types of solutions, we can see that the relationships between slopes and y-intercepts determine whether a system has one solution, no solution, or infinite solutions.