write the equation of a line parallel to y = -3x + 6, through (3,-4)

Answer 1

Hi Kendall!

A parallel line means our new line will have a slope equal to the original line.

The original line is in slope-intercept form, y = mx + b, where m is the slope and b is the value of the y-intercept. This tells us that the slope we want to use is m = -3

By using a slope of -3, with the given point (3,-4), we can now use the point-slope formula to find our new line.

Point-slope formula: y - y1 = m(x - x1) where m is the slope and the line passes through the point (x1,y1)

Combining our point and slope, we get:

y + 4 = -3(x - 3)

If we wish, we can rearrange this by distributing on the right hand side.

y + 4 = -3x + 9

We can then subtract 4 from both sides to get our answer in slope-intercept form

y + 4 - 4 = -3x + 9 - 4

y = -3x + 5

Answer 2

It's our favorite, the slope intercept form of a linear equation, y = mx + b where m = gradient, and b = y intercept.

If your equation is not in this form, you must rearrange. But it already is, so less work for us. :-)

If y = -3x + 6 then m = -3 which tells us the slope and b = 6 which tells us it passes through (0,6)

For parallel lines, the gradients are always the same. So the general equation for (infinite) parallel (to our given line) lines is y = -3x + n.

But we need the specific value of n which makes the line pass through (3,-4).

Let's call this line y = -3x + c and plug in the point values to get -4 = -3(3) + c

=> -4 = -9 + c

=> c = 5

Hence our required equation is y = -3x + 5 OR the equivalent y = 5 - 3x.

Check out Desmos graphing calculator if you want to see what this looks like.