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

To find the value of \(2a + b\), we need to determine the values of \(a\) and \(b\) from the given linear equation \(a y = x + b\) which passes through the points \((-5, -2)\) and \((4, 7)\).

1. First, we convert the equation \(a y = x + b\) into slope-intercept form.
   • Rearrange it to: \(y = \frac{1}{a}x + \frac{b}{a}\).
   • This form \(y = mx + c\) suggests that the slope \(m = \frac{1}{a}\).
2. Calculate the slope \(m\) using the two points:
   • Formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
   • Plug in the values: \(m = \frac{7 - (-2)}{4 - (-5)} = \frac{7 + 2}{4 + 5} = \frac{9}{9} = 1\).
   • Thus, \(m = 1\), so \(\frac{1}{a} = 1\), which implies \(a = 1\).
3. Substitute the value of \(a\) into one of the points to find \(b\).
   • Using the equation \(y = x + b\) and the point \((4, 7)\), substitute \(a = 1\):
   • \(7 = 4 + b\).
   • Solve for \(b\): \(b = 7 - 4 = 3\).
4. Now, calculate \(2a + b\):
   • Substitute \(a = 1\) and \(b = 3\) into the expression:
   • \(2a + b = 2 \times 1 + 3 = 2 + 3 = 5\).

Hence, the value of \(2a + b\) is 5.

Therefore, the correct option is 5.