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
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.
YOU NEED TO SOLVE
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.