The equation of a line is given by 𝑦=𝑎(𝑥−5)+4.If the 𝑦-intercept is −6, find the value of 𝑎.

1. Break Down the Problem

We have the equation of the line $y = a(x - 5) + 4$ and we need to find the value of $a$ given that the $y$-intercept is $-6$. The $y$-intercept occurs when $x = 0$.

2. Relevant Concepts

The $y$-intercept can be found by substituting $x = 0$ into the equation of the line. The point of intersection with the $y$-axis is given by the coordinates $(0, c)$, where $c$ is the $y$-intercept.

3. Analysis and Detail

Substituting $x = 0$ into the equation:

$y = a(0 - 5) + 4$

This simplifies to:

$y = a(-5) + 4$ $y = -5a + 4$

Since we know the $y$-intercept is $-6$, we can set up the following equation:

$-5a + 4 = -6$

4. Verify and Summarize

Now, we will solve for $a$:

$-5a + 4 = -6 \\ -5a = -6 - 4 \\ -5a = -10 \\ a = \frac{-10}{-5} \\ a = 2$

Final Answer

The value of $a$ is $2$.