Questions: Now, find the equation in terms of x for a line that passes through the two points (-1,3) and (4,-5). Express your answer in terms of x. The " y= " has been given for you.

Now, find the equation in terms of x for a line that passes through the two points (-1,3) and (4,-5). Express your answer in terms of x. The " y= " has been given for you.

Transcript text: Now, find the equation in terms of $x$ for a line that passes through the two points $(-1,3)$ and $(4,-5)$. Express your answer in terms of $x$. The " $y=$ " has been given for you.

Solution

Solution Steps

To find the equation of a line that passes through two given points, we need to determine the slope of the line and then use the point-slope form of the equation of a line. The slope $ m $ is calculated as the change in $ y $ divided by the change in $ x $. Once we have the slope, we can use one of the points to find the y-intercept $ b $ using the equation $ y = mx + b $.

Solution Approach

Calculate the slope $ m $ using the formula $ m = \frac{y_2 - y_1}{x_2 - x_1} $.
Use the slope and one of the points to solve for the y-intercept $ b $ using the equation $ y = mx + b $.
Write the final equation in the form $ y = mx + b $.

Step 1: Calculate the Slope

To find the equation of the line passing through the points $(-1, 3)$ and $ (4, -5) $, we first calculate the slope $ m $. The slope is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-5 - 3}{4 - (-1)} = \frac{-8}{5} = -1.6 \]

Step 2: Calculate the Y-Intercept

Next, we use the slope $ m $ and one of the points to find the y-intercept $ b $. Using the point $(-1, 3)$: \[ y = mx + b \implies 3 = -1.6(-1) + b \implies 3 = 1.6 + b \implies b = 3 - 1.6 = 1.4 \]