Questions: Find a formula for the linear function with a slope of -4 and an x-intercept value of 5 : y=

Transcript text: Find a formula for the linear function with a slope of -4 and an $x$-intercept value of 5 : \[ y= \] $\square$

Solution

Solution Steps

To find the formula for a linear function given the slope and the x-intercept, we can use the point-slope form of a linear equation. The point-slope form is $ y - y_1 = m(x - x_1) $, where $ m $ is the slope and $ (x_1, y_1) $ is a point on the line. Since the x-intercept is given, we know that $ y = 0 $ when $ x = 5 $. We can substitute these values into the point-slope form to find the equation.

Solution Approach

Use the point-slope form of the linear equation.
Substitute the given slope and the x-intercept into the equation.
Simplify the equation to get it into the slope-intercept form $ y = mx + b $.

Step 1: Identify the Given Values

We are given the slope $ m = -4 $ and the $ x $-intercept $ x = 5 $.

Step 2: Use the Point-Slope Form

The point-slope form of a linear equation is: \[ y - y_1 = m(x - x_1) \] Since the $ x $-intercept is 5, we know that $ y = 0 $ when $ x = 5 $. Therefore, $ (x_1, y_1) = (5, 0) $.

Step 3: Substitute the Given Values

Substitute $ m = -4 $, $ x_1 = 5 $, and $ y_1 = 0 $ into the point-slope form: \[ y - 0 = -4(x - 5) \]

Step 4: Simplify to Slope-Intercept Form

Simplify the equation to get it into the slope-intercept form $ y = mx + b $: \[ y = -4(x - 5) \] \[ y = -4x + 20 \]