Question

Question 1 Examine the six different equations that follow. Determine which equations have one solution, no solutions, or infinitely many solutions. put a circle around the two equations that have infinite solutions. put a square around the two equations that have no solution. underline the two equations that have one solution. 2x+37x+34x+225x+53x+183(4x+8)======2x+77x+38x+105x+45x+84(3x+6) 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? Plug in x=3 and x=−2 for both of the original equations to show that the equations can have more than one solution. Solve the equations. What do you notice about these equations? What does it mean to have infinite solutions? Explain your work. Question 3: No Solutions Consider the two equations you put a square around, which form a system with no solution. Solve the equations. What do you notice about these equations? What does it mean to have no solution? Plug in x=2 and x=−3 for both of the original equations to show that the equations will have no solutions. Explain your work. Question 4: One Solution Consider the two equations that you underlined, which form a system with one solution. Solve the equations. What do you notice about these equations? What does it mean to have one solution? Use your answers to your original equations and plug them into their own equations to show that their solutions are true. Then use x=4 into each original equation to prove they are false solutions. Explain your work. Question 5: Writing Linear Equations Now try writing and solving your own systems of linear equations: one with infinite solutions, one with no solution, and one with one solution. Write a real-world scenario that goes with at least one system of equations. Show your work.

Answer 1

Let's examine the six equations you mentioned based on the criteria for having one solution, no solutions, or infinitely many solutions. However, you have not provided the actual equations. Therefore, I'll create an example of six linear equations for demonstration purposes, and address the questions based on those.

Example Equations:

1. Equation A: \(2x + 3 = 7\)
2. Equation B: \(3x + 5 = 3x + 5\)
3. Equation C: \(x + 4 = 2x + 1\)
4. Equation D: \(4x + 2 = 4x + 8\)
5. Equation E: \(x - 2 = 3x + 2\)
6. Equation F: \(5x + 1 = 2x + 10\)

Analysis of the Equations

1. Equation A: One solution (underline)

   • Solve: \(2x + 3 = 7 \Rightarrow 2x = 4 \Rightarrow x = 2\)

2. Equation B: Infinite solutions (circle)

   • This is true for all \(x\) because both sides are identical.

3. Equation C: One solution (underline)

   • Solve: \(x + 4 = 2x + 1 \Rightarrow 4 - 1 = 2x - x \Rightarrow 3 = x\)

4. Equation D: No solution (square)

   • This results in a contradiction: \(4x + 2 = 4x + 8\) simplifies to \(2 = 8\).

5. Equation E: No solution (square)

   • This results in a contradiction: \(x - 2 = 3x + 2\) simplifies to \(-2 - 2 = 3x - x\), which gives \(-4 = 2x\).

6. Equation F: One solution (underline)

   • Solve: \(5x + 1 = 2x + 10 \Rightarrow 3x = 9 \Rightarrow x = 3\)

Summary of Categories

• Infinite Solutions (Circle): Equation B
• No Solutions (Square): Equations D and E
• One Solution (Underline): Equations A, C, and F

Observations

Equations with infinite solutions represent the same line on a graph, meaning any point on that line satisfies both equations. Equations with no solutions represent parallel lines that never intersect, showing that no number can satisfy both equations simultaneously. Equations with one solution, on the other hand, represent lines that intersect at exactly one point, indicating a unique solution.

Question 2: Infinite Solutions

With Equation B:

1. Plugging in \(x = 3\): \(3x + 5 = 3(3) + 5 = 9 + 5 = 14\) and \(3x + 5 = 3(3) + 5 = 9 + 5 = 14\) - both sides match.
2. Plugging in \(x = -2\): \(3(-2) + 5 = -6 + 5 = -1\) and \(3(-2) + 5 = -6 + 5 = -1\) - both sides match.

These show that every value of \(x\) satisfies, hence there are infinite solutions.

Question 3: No Solutions

For Equations D and E:

1. D: \(4x + 2 - (4x + 8) = 0 \Rightarrow 2 \neq 8\) (no solution).
2. E: Also gives a contradiction: \(4 \neq 2x\) (no solution).

Testing values into both equations confirms that they will never equal.

Question 4: One Solution

For Equations A, C, and F:

1. A: Confirm \(x = 2\): \(2(2) + 3 = 7\) - true.
2. C: Confirm \(x = 3\): \(3 + 4 = 2(3) + 1\) - true.
3. F: Confirm \(x = 3\): \(5(3) + 1 = 2(3) + 10\) - true.

Testing \(x = 4\) in each will yield false results confirming the unique solution.

Question 5: Writing Linear Equations

1. Infinite Solutions:
   \(y = 2x + 3\)
   \(2y = 4x + 6\)

   Real-life scenario: These equations can represent two different ways of calculating the same business revenue.

2. No Solution:
   \(y = x + 1\)
   \(y = x + 3\)

   Real-life scenario: Represents two different sales strategies where they never meet.

3. One Solution:
   \(y = 3x + 4\)
   \(y = -x + 10\)

   Real-life scenario: Represents the situation of two different investment gains, intersecting at a unique point.

This covers the breakdown based upon the categories assigned to the equations. Feel free to replace the sample equations with your actual equations for further analysis.