Question

If \( ay = x + b \) is the equation of the line passing through the points (-5, -2) and (4, 7), then the value of \( 2a + b \) is equal to:

• A. 1
• B. 3
• C. 5
• D. -3
• E. -1

Hint:

The equation of the line in slope-intercept form is useful for calculating the slope and intercept. The formula for the slope between two points is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

Answer 1

The Correct Option is C

The equation of the line is given as \( ay = x + b \), which we can rewrite as: \[ y = \frac{1}{a}x + \frac{b}{a} \] This is in the slope-intercept form, where \( \frac{1}{a} \) is the slope \( m \) of the line and \( \frac{b}{a} \) is the y-intercept. We are given two points on the line: (-5, -2) and (4, 7). Step 1: Find the slope.
The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the points (-5, -2) and (4, 7): \[ m = \frac{7 - (-2)}{4 - (-5)} = \frac{7 + 2}{4 + 5} = \frac{9}{9} = 1 \] So, the slope of the line is \( m = 1 \). From the equation \( y = \frac{1}{a}x + \frac{b}{a} \), we have \( \frac{1}{a} = 1 \), which gives \( a = 1 \).
Step 2: Find \( b \).
Now that we know \( a = 1 \), substitute one of the points into the equation to solve for \( b \). Using the point (4, 7): \[ 7 = \frac{1}{1} \times 4 + \frac{b}{1} \] \[ 7 = 4 + b \] \[ b = 7 - 4 = 3 \]
Step 3: Calculate \( 2a + b \).
Now, substitute \( a = 1 \) and \( b = 3 \) into \( 2a + b \): \[ 2a + b = 2(1) + 3 = 2 + 3 = 5 \]
Final Answer:} (C) 5