Joshua W. answered • 03/02/23
Down-to-Earth University Math Professor
What is an equation of the line that passes through the point ( − 2 , − 3 ) and is parallel to the line 4x-y=1
For an equation of a line,
y-y1=m(x-x1)
We need the slope m and a point that the line passes through (x1,y1), in order to construct the equation of said line.
We already know the point that it passes through, (-2,-3), so x1=-2 and y1=-3.
Now to find m, we don't have two points that the line passes through so we can't use the slope formula.
But we do know that lines that are parallel have equal slope. So if we find the slope of the line described by the equation 4x-y=1, then that is also the slope, or m, of the line we want.
To find the slope of the line described by the equation 4x-y=1, it helps to use algebra to convert it to slope-intercept form (y=mx+b), because then the slope can be read off as the number in front of the x.
Solving or isolating y in the equation 4x-y=1:
4x-y=1
subtract 4x from both sides
-y=-4x+1
divide both sides by -1
y=4x-1
Thus the slope of that given line is m=4, the number in front of the x. Now we have everything we want for the equation of a line. We have the slope=m=4, and the point that it goes through (-2,-3), so x1=-2 and y1=-3. Plugging it into the point-slope form equation of a line:
y-y1=m(x-x1)
y-(-3)=4(x-(-2))
y+3=4(x+2)
The above is in point-slope form for the equation of a line.
If you want to convert it to slope-intercept form, just distribute the 4 and subtract 3 to get the y by itself:
y+3=4(x+2)
y+3=4x+8
y=4x+8-3
y=4x+5 <-- answer they're looking for.
If you'd like to check your answer,
the slope is number in front of the x, so the slope is m=4. Does it pass through the point (-2,-3)? Let's plug in x=-2 and y=-3 to see if it satisfies the equation:
y=4x+5
-3=4(-2)+5
-3=-8+5
-3=-3. Check.
So the equation y=4x+5 has the same slope as 4x-y=1 (m=4) and passes through the point (-2,-3).
