How do you write the equation in slope intercept form parallel to y = 3x+6 and passes through the point (-10, 2.5)?

Question

Nathan Axel · Accepted Answer

It is #y=3x+32.5#.

What makes two lines parallels is the slope. If the slope is the same, the lines cannot intersect. Let's try to see it algebraically. We have the first line with the equation #y=mx+a# and the second line has the same slope #m# so has equation #y=mx+b# If the lines have an intersection it means that the #x# and the #y# are the same. So we can, for example, equate the #y# and say #mx+a = mx+b# because they both are equal to the same #y#. But we can remove the #mx# from both equations because it is identical and we obtain #a=b#. What does it mean? Clearly if #a=b# the two lines are the same line, and this is the only case when two parallel line can have an intersection, when they intersects everywhere (and are then the same line). So if #a# is not equal to #b# the lines cannot intersect. Now that we know how to describe a parallel line, we have #y=3x+6#. All the line with the form #y=3x+a# are parallel. So #y=3x+2# is parallel, #y=3x-18# is parallel etc. There are infinite parallels, one for each chose of #a#. We are searching for one in particular, it is the line that passes from the point #(-10, 2.5)#. We then know that when #x=-10# we must have #y=2.5#. We use this information in our general equation of the parallel #y=3x+a#. #y=3x+a# and #x=-10, y=2.5#, then #2.5=3*(-10)+a# #2.5=-30+a# and we can solve for #a# #a=32.5#. This fix one parallel that is #y=3x+32.5#.

Abigail Allen · Answer

undefined To write the equation of a line in slope-intercept form parallel to $ y = 3x + 6 $ and passing through the point $ (-10, 2.5) $, we first need to determine the slope of the given line, which is $ m = 3 $. Since parallel lines have the same slope, the slope of the new line will also be $ m = 3 $.

Using the point-slope form of a linear equation $ y - y_1 = m(x - x_1) $, where $ (x_1, y_1) $ is the given point and $ m $ is the slope, we substitute the given values:

$ y - 2.5 = 3(x + 10) $

Next, we can simplify and rewrite the equation in slope-intercept form $ y = mx + b $ by solving for $ y $:

$ y - 2.5 = 3x + 30 $

$ y = 3x + 30 + 2.5 $

$ y = 3x + 32.5 $

Therefore, the equation of the line parallel to $ y = 3x + 6 $ and passing through the point $ (-10, 2.5) $ in slope-intercept form is $ y = 3x + 32.5 $.