Questions: Find the equation (in terms of (x)) of the line through the points ((-3,1)) and ((2,-4)) (y=)

Transcript text: Find the equation (in terms of $x$ ) of the line through the points $(-3,1)$ and $(2,-4)$ $y=$ $\square$

Solution

Solution Steps

To find the equation of the line through two points, we need to determine the slope and then use the point-slope form of a line. First, calculate the slope using the formula $(y_2 - y_1) / (x_2 - x_1)$. Then, use the point-slope form $y - y_1 = m(x - x_1)$ to find the equation of the line, where $m$ is the slope and $(x_1, y_1)$ is one of the given points.

Step 1: Calculate the Slope

To find the slope $ m $ of the line passing through the points $(-3, 1)$ and $(2, -4)$, we use the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 1}{2 - (-3)} = \frac{-5}{5} = -1.0 \]

Step 2: Calculate the Y-Intercept

Next, we use the point-slope form of the line to find the y-intercept $ b $. The point-slope form is given by:

\[ y - y_1 = m(x - x_1) \]

Substituting one of the points, say $(-3, 1)$, into the equation:

\[ y - 1 = -1.0(x - (-3)) \]

Rearranging to find $ b $:

\[ y - 1 = -1.0(x + 3) \implies y = -1.0x - 3 + 1 \implies y = -1.0x - 2.0 \]

Thus, the y-intercept $ b $ is $-2.0$.

Step 3: Write the Equation of the Line