Question

1. write an equation in point-slope form that has the same slope as (y = \frac{5}{4}x + 4) and passes through the point ((2, 5)).

(y - 2 = \frac{5}{4}(x - 5))

(y - 5 = 4(x - 2))

(y - 2 = 4(x - 5))

(y - 5 = \frac{5}{4}(x - 2))

Explanation:

🆕 New Concept Discovered: Point-Slope Form
Writing linear equations using a point and a slope.

Step 1: Identify the slope

The given equation is in slope-intercept form, \( y = mx + b \):

$$ y = \frac{5}{4}x + 4 $$

The slope \( m \) is the coefficient of \( x \):

$$ m = \frac{5}{4} $$

Step 2: Identify the point

Looking at the options, they all use the coordinates \( 2 \) and \( 5 \). Let's test the point \( (5, 2) \) with our slope \( m = \frac{5}{4} \).

Step 3: Write the equation in point-slope form

The point-slope form of a linear equation is:

$$ y - y_1 = m(x - x_1) $$

Substituting \( m = \frac{5}{4} \) and the point \( (x_1, y_1) = (5, 2) \):

$$ y - 2 = \frac{5}{4}(x - 5) $$

This matches the first option.

Answer:

\( y - 2 = \frac{5}{4}(x - 5) \)