Finding an equation of a line through $(4, -2)$ that is parallel to $-5x + 7y = -69$ in standard form

Question

Find an equation of a line satisfying the conditions given. Express the equation in standard form. Through $(4, -2)$ and parallel to $-5x+7y=-69$.

So I have already inserted the equation and coordinates into the point slope formula. But, I am unsure of what to do next. What is the next step in solving this problem?

Answer 1

In the point-slope formula $y-y_1 = m(x-x_1)$, you still need to find the slope $m$. The slope of a line in standard form is given by $-\frac{A}{B}$ in the equation $Ax+By=C$. This can be derived from solving for $y$ and finding the coefficient of $x$ in slope-intercept form.

Can you continue?

Answer 2

Alternatively, you can notice the coefficients $A$ and $B$ in the standard form $Ax+By=C$ uniquely determine the slope. Therefore when $(4,-2)$ is substituted, the left-hand side must equal the right-hand side.

This directly gives you the equation of a line in standard form.

Answer 3

Point-slope form is $y-y_1=m(x-x_1)$, where $(x_1,y_1)$ is a point on the line, and $m$ is the slope.

The slope of $-5x+7y=-69$ is $\dfrac57$. You could find this by, for instance, putting the equation in slope-intercept form: $y=mx+b$.

Now use the point $(4,-2)$, along with the slope, $\dfrac57$, to get $y-(-2)=\dfrac 57(x-4)$, or $y+2=\dfrac 57(x-4)$.

Finally, standard-form is $Ax+By=C$. So manipulate the above to get $\boxed{5x-7y=34}$.