Question

Examine the different equations that follow:

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. v = 5x

Complete steps a. through e. to determine which pair of equations form a system with one solution, no solution, or infinite solutions.

a. Solve each equation and make sure it is in y = mx + b format.

You need to solve EACH EQUATIONS

Answer 1

YOU NEED TO SOLVE

Answer 2

1. y = 5x - 4 (already in y = mx + b format)
2. y = -2x + 3 (already in y = mx + b format)
3. y = 3x + 1 (already in y = mx + b format)
4. y = 3x + 1 (divide by 2 to get y alone on one side)
5. y = 5x + 2 (add 5x to both sides)
6. y = 4x - 1 (add 4x and subtract 1 from both sides)
7. v = 5x (rewrite as y = 5x since we are comparing equations in y = mx + b format)


Equations in y = mx + b format:
1. y = 5x - 4
2. y = -2x + 3
3. y = 3x + 1
4. y = 3x + 1
5. y = 5x + 2
6. y = 4x - 1
7. y = 5x

b. Compare equations 1-3 to each other to determine if they have one solution, no solution, or infinite solutions.

Equation 1 (y = 5x - 4):
- Equation 2 (y = -2x + 3) has one solution as they have different slopes and y-intercepts.
- Equation 3 (y = 3x + 1) has one solution as they have different slopes and y-intercepts.

Therefore, equations 1, 2, and 3 have one solution.

c. Compare equation 4 with equations 1-3 to determine if they have one solution, no solution, or infinite solutions.

Equation 4 (y = 3x + 1):
- Equation 1 (y = 5x - 4) has one solution as they have different slopes and y-intercepts.
- Equation 2 (y = -2x + 3) has one solution as they have different slopes and y-intercepts.
- Equation 3 (y = 3x + 1) has one solution as they have the same equation.

Therefore, equation 4 has one solution.

d. Compare equation 5 with equations 1-3 to determine if they have one solution, no solution, or infinite solutions.

Equation 5 (y = 5x + 2):
- Equation 1 (y = 5x - 4) has no solution as they have the same slope but different y-intercepts.
- Equation 2 (y = -2x + 3) has no solution as they have the same slope but different y-intercepts.
- Equation 3 (y = 3x + 1) has no solution as they have different slopes and y-intercepts.

Therefore, equation 5 has no solution.

e. Compare equation 6 with equations 1-3 to determine if they have one solution, no solution, or infinite solutions.

Equation 6 (y = 4x - 1):
- Equation 1 (y = 5x - 4) has one solution as they have different slopes and y-intercepts.
- Equation 2 (y = -2x + 3) has one solution as they have different slopes and y-intercepts.
- Equation 3 (y = 3x + 1) has one solution as they have different slopes and y-intercepts.

Therefore, equation 6 has one solution.

In conclusion, equations 1, 2, and 3 have one solution, equation 4 has one solution, equation 5 has no solution, and equation 6 has one solution.

Answer 3

I apologize for not solving the equations in the comparison steps. Let me solve them now:

a. Solve each equation and write them in y = mx + b format:

1. y = 5x - 4
2. y = -2x + 3
3. y = 3x + 1
4. y = 3x/2 + 1
5. y = 5x + 2
6. y = 4x + 1
7. y = 5x

Now that the equations are in y = mx + b format, we can compare them to determine if they have one solution, no solution, or infinite solutions.

b. Compare equations 1-3:

1. y = 5x - 4
2. y = -2x + 3
3. y = 3x + 1

Equations 1, 2, and 3 each have unique slopes and y-intercepts, indicating they each have one solution.

c. Compare equation 4 with equations 1-3:

4. y = 3x/2 + 1

Equation 4 has a different slope compared to equations 1-3, indicating one solution.

d. Compare equation 5 with equations 1-3:

5. y = 5x + 2

Equation 5 has the same slope as equation 1, but different y-intercept, suggesting no solution.

e. Compare equation 6 with equations 1-3:

6. y = 4x + 1

Equation 6 has a different slope compared to equations 1-3, indicating one solution.

Therefore, equations 1, 2, and 3 have one solution, equation 4 has one solution, equation 5 has no solution, and equation 6 has one solution. Thank you for pointing that out.