Answer :
To determine the equation of the line that passes through the points \( (1, 6) \) and \( (2, 1) \), we have to calculate the slope of the line and the y-intercept. Let’s go through this process step-by-step:
### Step 1: Calculate the Slope
The slope (\( m \)) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, \( (x_1, y_1) = (1, 6) \) and \( (x_2, y_2) = (2, 1) \). Thus, the slope \( m \) is:
[tex]\[ m = \frac{1 - 6}{2 - 1} = \frac{-5}{1} = -5 \][/tex]
### Step 2: Determine the Y-intercept
The equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
where \( m \) is the slope and \( b \) is the y-intercept.
We can use one of the points and the slope we calculated to find the y-intercept. Let’s use the point \( (1, 6) \).
Substitute \( m = -5 \), \( x = 1 \), and \( y = 6 \) into the equation:
[tex]\[ 6 = -5 \cdot 1 + b \][/tex]
Solving for \( b \):
[tex]\[ 6 = -5 + b \][/tex]
[tex]\[ b = 6 + 5 \][/tex]
[tex]\[ b = 11 \][/tex]
### Step 3: Form the Equation
Now that we have the slope and the y-intercept, we can write the equation of the line:
[tex]\[ y = -5x + 11 \][/tex]
### Step 4: Compare with Given Options
The equation we derived is \( y = -5x + 11 \). Comparing this with the given options:
A. \( y = -5x + 1 \)
B. \( y = -5x + 11 \)
C. \( y = 2x + 1 \)
D. \( y = 5x - 1 \)
The correct option is:
[tex]\[ \boxed{B} \][/tex]
### Step 1: Calculate the Slope
The slope (\( m \)) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Here, \( (x_1, y_1) = (1, 6) \) and \( (x_2, y_2) = (2, 1) \). Thus, the slope \( m \) is:
[tex]\[ m = \frac{1 - 6}{2 - 1} = \frac{-5}{1} = -5 \][/tex]
### Step 2: Determine the Y-intercept
The equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
where \( m \) is the slope and \( b \) is the y-intercept.
We can use one of the points and the slope we calculated to find the y-intercept. Let’s use the point \( (1, 6) \).
Substitute \( m = -5 \), \( x = 1 \), and \( y = 6 \) into the equation:
[tex]\[ 6 = -5 \cdot 1 + b \][/tex]
Solving for \( b \):
[tex]\[ 6 = -5 + b \][/tex]
[tex]\[ b = 6 + 5 \][/tex]
[tex]\[ b = 11 \][/tex]
### Step 3: Form the Equation
Now that we have the slope and the y-intercept, we can write the equation of the line:
[tex]\[ y = -5x + 11 \][/tex]
### Step 4: Compare with Given Options
The equation we derived is \( y = -5x + 11 \). Comparing this with the given options:
A. \( y = -5x + 1 \)
B. \( y = -5x + 11 \)
C. \( y = 2x + 1 \)
D. \( y = 5x - 1 \)
The correct option is:
[tex]\[ \boxed{B} \][/tex]
