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Sagot :
To determine which of the given equations represents the data table, we need to check each equation against the data points provided. The data points are:
[tex]\[ (0, 3), (2, 7), (4, 11), (6, 15) \][/tex]
Let's test each equation with these points to see which one fits all the data points.
### Option A: [tex]\( y = 2x - 3 \)[/tex]
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 2(0) - 3 = -3 \quad \text{(This does not match \( y = 3 \))} \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 2(2) - 3 = 4 - 3 = 1 \quad \text{(This does not match \( y = 7 \))} \][/tex]
3. For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 2(4) - 3 = 8 - 3 = 5 \quad \text{(This does not match \( y = 11 \))} \][/tex]
4. For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = 2(6) - 3 = 12 - 3 = 9 \quad \text{(This does not match \( y = 15 \))} \][/tex]
So, Option A does not match the data points.
### Option B: [tex]\( y = 4x - 1 \)[/tex]
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4(0) - 1 = -1 \quad \text{(This does not match \( y = 3 \))} \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 4(2) - 1 = 8 - 1 = 7 \quad \text{(This matches \( y = 7 \))} \][/tex]
3. For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 4(4) - 1 = 16 - 1 = 15 \quad \text{(This does not match \( y = 11 \))} \][/tex]
4. For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = 4(6) - 1 = 24 - 1 = 23 \quad \text{(This does not match \( y = 15 \))} \][/tex]
So, Option B does not match the data points.
### Option C: [tex]\( y = 2x + 3 \)[/tex]
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 2(0) + 3 = 3 \quad \text{(This matches \( y = 3 \))} \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 2(2) + 3 = 4 + 3 = 7 \quad \text{(This matches \( y = 7 \))} \][/tex]
3. For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 2(4) + 3 = 8 + 3 = 11 \quad \text{(This matches \( y = 11 \))} \][/tex]
4. For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = 2(6) + 3 = 12 + 3 = 15 \quad \text{(This matches \( y = 15 \))} \][/tex]
So, Option C matches all the data points.
### Option D: [tex]\( y = 4x + 3 \)[/tex]
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4(0) + 3 = 3 \quad \text{(This matches \( y = 3 \))} \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 4(2) + 3 = 8 + 3 = 11 \quad \text{(This does not match \( y = 7 \))} \][/tex]
3. For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 4(4) + 3 = 16 + 3 = 19 \quad \text{(This does not match \( y = 11 \))} \][/tex]
4. For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = 4(6) + 3 = 24 + 3 = 27 \quad \text{(This does not match \( y = 15 \))} \][/tex]
So, Option D does not match the data points.
### Conclusion
The equation [tex]\( y = 2x + 3 \)[/tex] (Option C) represents the data table correctly.
[tex]\[ (0, 3), (2, 7), (4, 11), (6, 15) \][/tex]
Let's test each equation with these points to see which one fits all the data points.
### Option A: [tex]\( y = 2x - 3 \)[/tex]
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 2(0) - 3 = -3 \quad \text{(This does not match \( y = 3 \))} \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 2(2) - 3 = 4 - 3 = 1 \quad \text{(This does not match \( y = 7 \))} \][/tex]
3. For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 2(4) - 3 = 8 - 3 = 5 \quad \text{(This does not match \( y = 11 \))} \][/tex]
4. For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = 2(6) - 3 = 12 - 3 = 9 \quad \text{(This does not match \( y = 15 \))} \][/tex]
So, Option A does not match the data points.
### Option B: [tex]\( y = 4x - 1 \)[/tex]
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4(0) - 1 = -1 \quad \text{(This does not match \( y = 3 \))} \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 4(2) - 1 = 8 - 1 = 7 \quad \text{(This matches \( y = 7 \))} \][/tex]
3. For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 4(4) - 1 = 16 - 1 = 15 \quad \text{(This does not match \( y = 11 \))} \][/tex]
4. For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = 4(6) - 1 = 24 - 1 = 23 \quad \text{(This does not match \( y = 15 \))} \][/tex]
So, Option B does not match the data points.
### Option C: [tex]\( y = 2x + 3 \)[/tex]
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 2(0) + 3 = 3 \quad \text{(This matches \( y = 3 \))} \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 2(2) + 3 = 4 + 3 = 7 \quad \text{(This matches \( y = 7 \))} \][/tex]
3. For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 2(4) + 3 = 8 + 3 = 11 \quad \text{(This matches \( y = 11 \))} \][/tex]
4. For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = 2(6) + 3 = 12 + 3 = 15 \quad \text{(This matches \( y = 15 \))} \][/tex]
So, Option C matches all the data points.
### Option D: [tex]\( y = 4x + 3 \)[/tex]
1. For [tex]\( x = 0 \)[/tex]:
[tex]\[ y = 4(0) + 3 = 3 \quad \text{(This matches \( y = 3 \))} \][/tex]
2. For [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 4(2) + 3 = 8 + 3 = 11 \quad \text{(This does not match \( y = 7 \))} \][/tex]
3. For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = 4(4) + 3 = 16 + 3 = 19 \quad \text{(This does not match \( y = 11 \))} \][/tex]
4. For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = 4(6) + 3 = 24 + 3 = 27 \quad \text{(This does not match \( y = 15 \))} \][/tex]
So, Option D does not match the data points.
### Conclusion
The equation [tex]\( y = 2x + 3 \)[/tex] (Option C) represents the data table correctly.
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