To find the linear equation of the line passing through the points [tex]\((2, -1)\)[/tex] and [tex]\((10, 7)\)[/tex], we will use the point-slope form of the equation of a line, which is given by:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Step 1: Identify the given points and compute the slope [tex]\( m \)[/tex]
The given points are:
- [tex]\( (x_1, y_1) = (2, -1) \)[/tex]
- [tex]\( (x_2, y_2) = (10, 7) \)[/tex]
The slope [tex]\( m \)[/tex] is calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points:
[tex]\[ m = \frac{7 - (-1)}{10 - 2} = \frac{7 + 1}{10 - 2} = \frac{8}{8} = 1.0 \][/tex]
Step 2: Use the point-slope form to find the equation
We know [tex]\( m = 1.0 \)[/tex] and one of the points is [tex]\((2, -1)\)[/tex]. Substituting these values into the point-slope form:
[tex]\[ y - (-1) = 1.0(x - 2) \][/tex]
[tex]\[ y + 1 = 1.0(x - 2) \][/tex]
Step 3: Convert to slope-intercept form [tex]\( y = mx + b \)[/tex]
Distribute the slope on the right-hand side:
[tex]\[ y + 1 = 1.0 \cdot (x - 2) \][/tex]
[tex]\[ y + 1 = 1.0 \cdot x - 1.0 \cdot 2 \][/tex]
[tex]\[ y + 1 = 1.0x - 2 \][/tex]
Isolate [tex]\( y \)[/tex] by subtracting 1 from both sides:
[tex]\[ y = 1.0x - 2 - 1 \][/tex]
[tex]\[ y = 1.0x - 3 \][/tex]
Therefore, the equation of the line in slope-intercept form is:
[tex]\[ y = 1.0x - 3.0 \][/tex]
Thus:
[tex]\[
y = \boxed{1.0} x + \boxed{-3.0}
\][/tex]
