Answer :
To solve the problem and determine which points are possible approximations for the given system of equations, we can follow these steps:
1. Examine the Equations:
We start with the two equations:
[tex]\[ y = -\frac{7}{4}x + \frac{5}{2} \][/tex]
[tex]\[ y = \frac{3}{4}x - 3 \][/tex]
2. Calculate the y-values for Each Given x:
We then calculate the y-values for each given x-coordinate in the points provided, based on both equations.
3. Check Proximity:
We compare if the given y-values in the points are close to the computed y-values from each equation. We consider points to be close enough if the difference is less than 0.1.
Let's evaluate the points one by one:
### Point (1.9, 2.5):
1. For Equation 1:
[tex]\[ y = -\frac{7}{4}(1.9) + \frac{5}{2} = -3.325 + 2.5 = -0.825 \][/tex]
2. For Equation 2:
[tex]\[ y = \frac{3}{4}(1.9) - 3 = 1.425 - 3 = -1.575 \][/tex]
The given y-value, 2.5, is not close to either -0.825 or -1.575. Thus, [tex]\((1.9, 2.5)\)[/tex] is not a good approximation.
### Point (2.2, -1.4):
1. For Equation 1:
[tex]\[ y = -\frac{7}{4}(2.2) + \frac{5}{2} = -3.85 + 2.5 = -1.35 \][/tex]
2. For Equation 2:
[tex]\[ y = \frac{3}{4}(2.2) - 3 = 1.65 - 3 = -1.35 \][/tex]
The given y-value, -1.4, is very close to -1.35 (difference is 0.05). Thus, [tex]\((2.2, -1.4)\)[/tex] is a reasonable approximation.
### Point (2.2, -1.35):
1. For Equation 1:
[tex]\[ y = -\frac{7}{4}(2.2) + \frac{5}{2} = -3.85 + 2.5 = -1.35 \][/tex]
2. For Equation 2:
[tex]\[ y = \frac{3}{4}(2.2) - 3 = 1.65 - 3 = -1.35 \][/tex]
The given y-value, -1.35, exactly matches the computed values. Thus, [tex]\((2.2, -1.35)\)[/tex] is a perfect approximation.
### Conclusion:
The points [tex]\((2.2, -1.4)\)[/tex] and [tex]\((2.2, -1.35)\)[/tex] are both possible approximations for the system of equations given. Therefore, we select these two options.
1. Examine the Equations:
We start with the two equations:
[tex]\[ y = -\frac{7}{4}x + \frac{5}{2} \][/tex]
[tex]\[ y = \frac{3}{4}x - 3 \][/tex]
2. Calculate the y-values for Each Given x:
We then calculate the y-values for each given x-coordinate in the points provided, based on both equations.
3. Check Proximity:
We compare if the given y-values in the points are close to the computed y-values from each equation. We consider points to be close enough if the difference is less than 0.1.
Let's evaluate the points one by one:
### Point (1.9, 2.5):
1. For Equation 1:
[tex]\[ y = -\frac{7}{4}(1.9) + \frac{5}{2} = -3.325 + 2.5 = -0.825 \][/tex]
2. For Equation 2:
[tex]\[ y = \frac{3}{4}(1.9) - 3 = 1.425 - 3 = -1.575 \][/tex]
The given y-value, 2.5, is not close to either -0.825 or -1.575. Thus, [tex]\((1.9, 2.5)\)[/tex] is not a good approximation.
### Point (2.2, -1.4):
1. For Equation 1:
[tex]\[ y = -\frac{7}{4}(2.2) + \frac{5}{2} = -3.85 + 2.5 = -1.35 \][/tex]
2. For Equation 2:
[tex]\[ y = \frac{3}{4}(2.2) - 3 = 1.65 - 3 = -1.35 \][/tex]
The given y-value, -1.4, is very close to -1.35 (difference is 0.05). Thus, [tex]\((2.2, -1.4)\)[/tex] is a reasonable approximation.
### Point (2.2, -1.35):
1. For Equation 1:
[tex]\[ y = -\frac{7}{4}(2.2) + \frac{5}{2} = -3.85 + 2.5 = -1.35 \][/tex]
2. For Equation 2:
[tex]\[ y = \frac{3}{4}(2.2) - 3 = 1.65 - 3 = -1.35 \][/tex]
The given y-value, -1.35, exactly matches the computed values. Thus, [tex]\((2.2, -1.35)\)[/tex] is a perfect approximation.
### Conclusion:
The points [tex]\((2.2, -1.4)\)[/tex] and [tex]\((2.2, -1.35)\)[/tex] are both possible approximations for the system of equations given. Therefore, we select these two options.
