Let's begin by addressing each part of the question in a step-by-step manner.
### Step 1: Find the product of the original equation and 4.
The original equation is:
[tex]\[ -5y + 9x = 11 \][/tex]
To find the product of this equation by 4, we multiply each term by 4:
[tex]\[ 4 \cdot (-5y) + 4 \cdot (9x) = 4 \cdot 11 \][/tex]
[tex]\[ -20y + 36x = 44 \][/tex]
So, the product of 4 and the linear equation is:
[tex]\[ \boxed{-20y + 36x = 44} \][/tex]
### Step 2: Write the original equation in slope-intercept form (y = mx + b).
The original equation is:
[tex]\[ -5y + 9x = 11 \][/tex]
To convert it to the slope-intercept form, we need to solve for \( y \):
[tex]\[ -5y = -9x + 11 \][/tex]
[tex]\[ y = \frac{-9x + 11}{-5} \][/tex]
[tex]\[ y = \frac{9}{5}x - \frac{11}{5} \][/tex]
So, the original equation in slope-intercept form is:
[tex]\[ \boxed{y = \frac{9}{5}x - \frac{11}{5}} \][/tex]
### Step 3: Write the new equation in slope-intercept form.
The new equation we obtained by multiplying the original equation by 4 is:
[tex]\[ -20y + 36x = 44 \][/tex]
Similarly, we solve for \( y \):
[tex]\[ -20y = -36x + 44 \][/tex]
[tex]\[ y = \frac{-36x + 44}{-20} \][/tex]
[tex]\[ y = \frac{36}{20}x - \frac{44}{20} \][/tex]
[tex]\[ y = \frac{9}{5}x - \frac{11}{5} \][/tex]
So, the new equation in slope-intercept form is:
[tex]\[ \boxed{y = \frac{9}{5}x - \frac{11}{5}} \][/tex]
### Summary
1. The product of 4 and the linear equation is: \( \boxed{-20y + 36x = 44} \).
2. The original equation in slope-intercept form is: \( \boxed{y = \frac{9}{5}x - \frac{11}{5}} \).
3. The new equation in slope-intercept form is: [tex]\( \boxed{y = \frac{9}{5}x - \frac{11}{5}} \)[/tex].
