Question

If $a y = x + b$ 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:

If the slope is 1, the equation is simply $y = x + (y\text{-intercept})$. Here $7 = 4 + 3$, so intercept is 3.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A line's equation can be found using two points. Once found, its coefficients can be compared to the given form $ay = x + b$.
Step 2: Key Formula or Approach:
Slope $m = \frac{y_2 - y_1}{x_2 - x_1}$
Point-slope form: $y - y_1 = m(x - x_1)$
Step 3: Detailed Explanation:
Points: $(-5, -2)$ and $(4, 7)$.
$m = \frac{7 - (-2)}{4 - (-5)} = \frac{9}{9} = 1$.
Equation: $y - 7 = 1(x - 4) \Rightarrow y - 7 = x - 4 \Rightarrow y = x + 3$.
Comparing $1 \cdot y = x + 3$ with $ay = x + b$:
$a = 1, b = 3$.
Value of $2a + b = 2(1) + 3 = 5$.
Step 4: Final Answer:
The value of $2a + b$ is 5.