π¦=5π₯β4
π¦=β2π₯+3
π¦=3π₯+1
2π¦=6π₯+2
π¦β5π₯=2
π¦β4π₯+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 π¦=ππ₯+π
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.
please someone help this is due today
so, what exactly have you done so far?
I see a fairly clear set of instructions.
I am sorry, as an AI language model, I can't see what the user has done so far. How can I assist you with the instructions?
To determine the nature of the solutions for each pair of equations, follow these steps:
Step a: Solve each equation and write it in y = mx + b format.
1. π¦ = 5π₯ β 4
Rewrite as y = 5x - 4.
2. π¦ = β2π₯ + 3
Rewrite as y = -2x + 3.
3. π¦ = 3π₯ + 1
Rewrite as y = 3x + 1.
4. 2π¦ = 6π₯ + 2
Divide both sides by 2 to get y = 3x + 1.
5. π¦ β 5π₯ = 2
Rewrite as y = 5x + 2.
6. π¦ β 4π₯ + 1 = 0
Rewrite as y = 4x - 1.
Step b: Graph all six equations on the coordinate grid and label them.
Step c: Circle the two equations that form a system with infinite solutions.
To identify the equations with infinite solutions, compare them to look for identical lines. In this case, equations 3 and 4 have the same equation, so circle those.
Step d: Put a square around the two equations that form a system with no solution.
To identify the equations with no solution, compare them to look for parallel lines. In this case, equations 5 and 6 have the same slope but different y-intercepts, so put a square around those.
Step e: Underline the two equations that form a system with one solution.
The remaining two equations, 1 and 2, form a system with one solution. Underline those.
Question 2: Infinite Solutions
Graph equations 3 and 4 together.
Observations:
- The graphs of equations 3 and 4 are the same line.
- A system having infinite solutions means that both equations represent the same line or overlap.
- Graphically, this means that the two lines coincide and all points on one line are also points on the other line.
To check that the system has infinite solutions, substitute points into both equations and verify that they are true for both equations.
Question 3: No Solution
Graph equations 5 and 6 together.
Observations:
- The graphs of equations 5 and 6 are parallel lines that never intersect.
- A system having no solution means that the two lines do not intersect.
- Graphically, this means that there is no point of intersection between the two lines.
To check that the system has no solution, substitute points into both equations and verify that they are not true for both equations.
Question 4: One Solution
Graph equations 1 and 2 together.
Observations:
- The graphs of equations 1 and 2 are two distinct lines that intersect at a single point.
- A system having one solution means that the two lines intersect at exactly one point.
- Graphically, this means that there is a unique point of intersection between the two lines.
To check that the system has one solution, substitute points into both equations and verify that they are true for both equations.