To complete the linear equation that represents the given table's data, we'll follow these steps:
1. Identify Two Points from the Table:
Let's select the first two points:
- Point 1: [tex]\((3, 82)\)[/tex]
- Point 2: [tex]\((4, 84)\)[/tex]
2. Calculate the Slope (m):
The formula to find the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
Substituting the coordinates of the selected points:
[tex]\[
m = \frac{84 - 82}{4 - 3} = \frac{2}{1} = 2.0
\][/tex]
3. Determine the Y-Intercept (b):
The equation of the line in slope-intercept form is [tex]\(y = mx + b\)[/tex]. To find [tex]\(b\)[/tex], we use one of the points and the slope.
Using the point [tex]\((3, 82)\)[/tex] and [tex]\(m = 2.0\)[/tex]:
[tex]\[
82 = 2.0 \cdot 3 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
82 = 6 + b \implies b = 82 - 6 = 76.0
\][/tex]
4. Write the Final Equation:
Substituting the values of [tex]\(m\)[/tex] and [tex]\(b\)[/tex] into the equation [tex]\(y = mx + b\)[/tex], we get:
[tex]\[
y = 2.0x + 76.0
\][/tex]
So, the filled-in linear equation is:
[tex]\[
y = 2.0x + 76.0
\][/tex]
