To determine which linear equation corresponds to the provided table of values, let's analyze each option in relation to the points given:
The table shows the following points:
- (0, 3)
- (3, 15)
- (7, 31)
- (10, 43)
We will evaluate each option:
Option A: [tex]\(y = 4x - 3\)[/tex]
1. For [tex]\(x = 0: y = 4(0) - 3 = -3\)[/tex] (this does not match [tex]\(y = 3\)[/tex])
2. Since the first point does not satisfy the equation, this option is incorrect.
Option B: [tex]\(y = \frac{1}{4}x - 3\)[/tex]
1. For [tex]\(x = 0: y = \frac{1}{4}(0) - 3 = -3\)[/tex] (this does not match [tex]\(y = 3\)[/tex])
2. Since the first point does not satisfy the equation, this option is also incorrect.
Option C: [tex]\(y = \frac{1}{4}x + 3\)[/tex]
1. For [tex]\(x = 0: y = \frac{1}{4}(0) + 3 = 3\)[/tex] (this matches the first point)
2. For [tex]\(x = 3: y = \frac{1}{4}(3) + 3 = \frac{3}{4} + 3 = 3.75\)[/tex] (this does not match [tex]\(y = 15\)[/tex])
Since the second point does not satisfy the equation, this option is incorrect.
Option D: [tex]\(y = 4x + 3\)[/tex]
1. For [tex]\(x = 0: y = 4(0) + 3 = 3\)[/tex] (this matches the first point)
2. For [tex]\(x = 3: y = 4(3) + 3 = 12 + 3 = 15\)[/tex] (this matches the second point)
3. For [tex]\(x = 7: y = 4(7) + 3 = 28 + 3 = 31\)[/tex] (this matches the third point)
4. For [tex]\(x = 10: y = 4(10) + 3 = 40 + 3 = 43\)[/tex] (this matches the fourth point)
Since all points given in the table satisfy the equation [tex]\(y = 4x + 3\)[/tex], this option is correct.
Therefore, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
