Respuesta :
Solution:
Given:
[tex]\begin{gathered} Li\text{ne 1;} \\ 3x+4y=12 \\ \text{Point (0,4)} \end{gathered}[/tex]To get the equation of the second line, it has the same slope as Line 1 because the two lines are parallel.
Hence, the slope
[tex]m_1=m_2[/tex][tex]\begin{gathered} 3x+4y=12 \\ In\text{ slope-intercept form,} \\ y=mx+b \\ m\text{ is the slope} \\ b\text{ is the y-intercept} \\ \\ \text{Thus;} \\ 3x+4y=12 \\ 4y=-3x+12 \\ \text{Dividing all through by 4,} \\ y=-\frac{3}{4}x+\frac{12}{4} \\ y=-\frac{3}{4}x+3 \\ \\ \text{Hence,} \\ m_1=-\frac{3}{4} \end{gathered}[/tex]To get the equation of the second line parallel to line 1, then;
[tex]\begin{gathered} m_1=m_2 \\ \text{Thus,} \\ m_2=-\frac{3}{4} \end{gathered}[/tex]The equation of the second line is gotten by the formula;
[tex]\begin{gathered} \frac{y-y_1}{x-x_1}=m \\ \text{where;} \\ x_1=0 \\ y_1=4 \\ m=-\frac{3}{4} \end{gathered}[/tex]Thus;
[tex]\begin{gathered} \frac{y-y_1}{x-x_1}=m \\ \frac{y-4}{x-0}=-\frac{3}{4} \\ \frac{y-4}{x}=-\frac{3}{4} \\ \text{Cross multiplying;} \\ 4(y-4)=-3x \\ 4y-16=-3x \\ 4y=-3x+16 \\ \text{Dividing both sides all through by 4 to get the equation in slope-intercept form;} \\ y=-\frac{3}{4}x+\frac{16}{4} \\ y=-\frac{3}{4}x+4 \end{gathered}[/tex]
Therefore, in slope-intercept form, the equation of the line parallel to 3x + 4y = 12 through the point (0,4) is;
[tex]y=-\frac{3}{4}x+4[/tex]