To find the equation of the line passing through the points [tex]\((2, 1)\)[/tex] and [tex]\((5, 10)\)[/tex], we need to determine the slope [tex]\(m\)[/tex] and the y-intercept [tex]\(b\)[/tex] for the line equation in the form [tex]\(y = mx + b\)[/tex].
### Step 1: Calculate the slope [tex]\(m\)[/tex]
The formula for 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]
Given points are [tex]\((2, 1)\)[/tex] and [tex]\((5, 10)\)[/tex]. Using these points:
[tex]\[ x_1 = 2, y_1 = 1 \][/tex]
[tex]\[ x_2 = 5, y_2 = 10 \][/tex]
Substitute the values into the slope formula:
[tex]\[ m = \frac{10 - 1}{5 - 2} \][/tex]
[tex]\[ m = \frac{9}{3} \][/tex]
[tex]\[ m = 3.0 \][/tex]
### Step 2: Determine the y-intercept [tex]\(b\)[/tex]
Once we have the slope, we can find the y-intercept [tex]\(b\)[/tex] using the point-slope form of the line equation. We choose one of the points, say [tex]\((2, 1)\)[/tex], and use the slope we just calculated.
The general form of the line equation is:
[tex]\[ y = mx + b \][/tex]
To find [tex]\(b\)[/tex], substitute [tex]\(m = 3\)[/tex], [tex]\(x = 2\)[/tex], and [tex]\(y = 1\)[/tex] into the equation:
[tex]\[ 1 = 3(2) + b \][/tex]
[tex]\[ 1 = 6 + b \][/tex]
[tex]\[ b = 1 - 6 \][/tex]
[tex]\[ b = -5.0 \][/tex]
### Final Equation of the Line
Combining the slope [tex]\(m = 3.0\)[/tex] and y-intercept [tex]\(b = -5.0\)[/tex], we get the equation of the line:
[tex]\[ y = 3.0x - 5.0 \][/tex]
Thus, the equation of the line passing through the points [tex]\((2, 1)\)[/tex] and [tex]\((5, 10)\)[/tex] is:
[tex]\[ y = 3.0x - 5.0 \][/tex]
