Write the equation of the line passing through (−3, 4) and perpendicular to −2xx + 7yy = −3

we want to clear for [tex]y[/tex] to get a slope-intercept equation ([tex]y=mx+b[/tex], where [tex]m[/tex] is the slope and [tex]b[/tex] is the interception of the line with the y axis)

[tex]7y=2x-3\\y=\frac{2}{7}x-\frac{3}{7}[/tex]

this way we can see that the number that represents the slope is [tex]\frac{2}{7}[/tex]. I will call this [tex]m_{1}=\frac{2}{7}[/tex] because it is the slope of the fisrt line.

Now to find the equation of the second line (the line perpendicular to [tex]-2x+7y=-3[/tex]), We need to apply the condition so that two lines are perpendicular:

[tex]m_{1}m_{2}=-1[/tex]

we have [tex]m_{1}=\frac{2}{7}[/tex], so the slope of the perpedicular line [tex]m_{2}[/tex] is:

[tex]\frac{2}{7}m_{2}=-1\\ m_{2}=-\frac{7}{2}[/tex]

we already have the slope, and the problem mentions that the new line passes through the point (-3, 4), so we use the point-slope equation

[tex]y=m(x-x_{0})+y_{0}[/tex]

where [tex]m[/tex] is the slope, and [tex](x_{0},y_{0})[/tex] is the point, so [tex]x_{0}=-3[/tex], and [tex]y_{0}=4[/tex]

thus:

[tex]y=-3.5(x-(-3))+4 \\y=-3.5(x+3)+4\\y=-3.5x-10.5+4\\y=-3.5x-6.5[/tex]

the equation is [tex]y=-3.5x-6.5[/tex]