To determine which linear equation represents the data chart accurately, we need to find the slope and the y-intercept of the line passing through the given points (1, 8), (2, 11), (3, 14), and (4, 17).
Let's start with the slope ([tex]\(m\)[/tex]) calculation:
[tex]\[ m = \frac{{y_2 - y_1}}{{x_2 - x_1}} \][/tex]
Using the first two points, [tex]\((x_1, y_1) = (1, 8)\)[/tex] and [tex]\((x_2, y_2) = (2, 11)\)[/tex]:
[tex]\[
m = \frac{{11 - 8}}{{2 - 1}} = \frac{3}{1} = 3
\][/tex]
So, the slope of the line is [tex]\(3\)[/tex].
Next, we need to find the y-intercept ([tex]\(b\)[/tex]) using the formula [tex]\( y = mx + b \)[/tex]:
Using the point [tex]\((1, 8)\)[/tex] and the slope [tex]\(3\)[/tex]:
[tex]\[
8 = 3 \cdot 1 + b \implies 8 = 3 + b \implies b = 8 - 3 = 5
\][/tex]
Therefore, the equation of the line is:
[tex]\[ y = 3x + 5 \][/tex]
Now, let's compare this with the given choices:
1. [tex]\( y = 3x + 5 \)[/tex]
2. [tex]\( y = x - 5 \)[/tex]
3. [tex]\( y = 3x + 11 \)[/tex]
Comparing our derived equation [tex]\( y = 3x + 5 \)[/tex] with the options given, we see that the correct equation is:
[tex]\[ y = 3x + 5 \][/tex]
Therefore, the linear equation that represents the data chart is:
[tex]\[ \boxed{y = 3x + 5} \][/tex]
