To find the equation of the line passing through the points [tex]\((5, -3)\)[/tex] and [tex]\((6, -1)\)[/tex], we will follow these steps:
### Step 1: Calculate the Slope (m)
The formula for the slope [tex]\(m\)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Given the points [tex]\((5, -3)\)[/tex] and [tex]\((6, -1)\)[/tex]:
[tex]\[ x_1 = 5, y_1 = -3 \][/tex]
[tex]\[ x_2 = 6, y_2 = -1 \][/tex]
Substitute these values into the slope formula:
[tex]\[ m = \frac{-1 - (-3)}{6 - 5} \][/tex]
[tex]\[ m = \frac{-1 + 3}{6 - 5} \][/tex]
[tex]\[ m = \frac{2}{1} \][/tex]
[tex]\[ m = 2 \][/tex]
### Step 2: Calculate the Y-Intercept (b)
We know the slope [tex]\(m = 2\)[/tex]. We use the slope-intercept form of the line equation [tex]\(y = mx + b\)[/tex]. To find the [tex]\(y\)[/tex]-intercept [tex]\(b\)[/tex], we can substitute one of the given points and the slope into this equation.
Using the point [tex]\((5, -3)\)[/tex]:
[tex]\[ y_1 = mx_1 + b \][/tex]
[tex]\[ -3 = 2 \cdot 5 + b \][/tex]
[tex]\[ -3 = 10 + b \][/tex]
Subtract 10 from both sides to solve for [tex]\(b\)[/tex]:
[tex]\[ -3 - 10 = b \][/tex]
[tex]\[ b = -13 \][/tex]
### Step 3: Write the Equation of the Line
We have determined the slope [tex]\(m = 2\)[/tex] and the [tex]\(y\)[/tex]-intercept [tex]\(b = -13\)[/tex]. Therefore, the equation of the line in slope-intercept form [tex]\(y = mx + b\)[/tex] is:
[tex]\[ y = 2x - 13 \][/tex]
Thus, the equation of the line that passes through the points [tex]\((5, -3)\)[/tex] and [tex]\((6, -1)\)[/tex] is:
[tex]\[ y = 2x - 13 \][/tex]
