SOLUTION: Can someone help me with this one. Determine whether the graphs of the equations are parallel lines, perpendicular lines or niether. 3x - 8y = -18 32x + 12y = -18 Thank you!

Algebra ->  Linear-equations -> SOLUTION: Can someone help me with this one. Determine whether the graphs of the equations are parallel lines, perpendicular lines or niether. 3x - 8y = -18 32x + 12y = -18 Thank you!      Log On


   

Question 158073: Can someone help me with this one.
Determine whether the graphs of the equations are parallel lines, perpendicular lines or niether.
3x - 8y = -18
32x + 12y = -18
Thank you!
Found 3 solutions by jim_thompson5910, Earlsdon, Electrified_Levi:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

3x-8y=-18 Start with the first equation.


-8y=-18-3x Subtract 3x from both sides.


-8y=-3x-18 Rearrange the terms.


y=%28-3x-18%29%2F%28-8%29 Divide both sides by -8 to isolate y.


y=%28%28-3%29%2F%28-8%29%29x%2B%28-18%29%2F%28-8%29 Break up the fraction.


y=%283%2F8%29x%2B9%2F4 Reduce.


So we can see that the equation y=%283%2F8%29x%2B9%2F4 has a slope m=3%2F8 and a y-intercept b=9%2F4.


32x%2B12y=-18 Now move onto the second equation


12y=-18-32x Subtract 32x from both sides.


12y=-32x-18 Rearrange the terms.


y=%28-32x-18%29%2F%2812%29 Divide both sides by 12 to isolate y.


y=%28%28-32%29%2F%2812%29%29x%2B%28-18%29%2F%2812%29 Break up the fraction.


y=-%288%2F3%29x-3%2F2 Reduce.


So we can see that the equation y=-%288%2F3%29x-3%2F2 has a slope m=-8%2F3 and a y-intercept b=-3%2F2.


So the slope of the first line is m=3%2F8 and the slope of the second line is m=-8%2F3.


Notice how the slope of the second line m=-8%2F3 is simply the negative reciprocal of the slope of the first line m=3%2F8.


In other words, if you flip the fraction of the second slope and change its sign, you'll get the first slope (or vice versa). So this means that y=%283%2F8%29x%2B9%2F4 and y=-%288%2F3%29x-3%2F2 are perpendicular lines.


Answer by Earlsdon(6294) About Me  (Show Source):
You can put this solution on YOUR website!
To determine whether the graphs of the given equations are perpendicular lines, parallel lines, or neither, you need ony to compare their slopes.
Perpendicular lines have negative reciprocal slopes.
Parallel lines have equal slopes, and neither have neither!
So, let's get the equations into their "slope-intercept" forms: y=mx+b.
1) 3x-8y+=+-18 Add 8y to both sides.
3x+=+8y-18 Add 18 to both sides.
3x%2B18+=+8y Divide both sides by 8.
highlight%28y+=+%283%2F8%29x%2B9%2F4%29 The slope here is:highlight%28m%5B1%5D+=+3%2F8%29
2) 32x%2B12y+=+-18 Subtract 2x from both sides.
12y+=+-32x-18 Divide both sides by 12.
highlight%28y+=+%28-8%2F3%29x-9%2F6%29 The slope here is: highlight%28m%5B2%5D+=+-8%2F3%29
Comparing m%5B1%5D+=+3%2F8 with m%5B2%5D+=+-8%2F3, you can see that they are negative reciprocals and, therefore, their graphs of the equations will be perpendicular lines.
For confirmation, we'll graph the two equations. Red = equation 1). Green = equation 2).

Answer by Electrified_Levi(103) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, Hope I can help,
.
Can someone help me with this one.
Determine whether the graphs of the equations are parallel lines, perpendicular lines or niether.
+3x+-+8y+=+%28-18%29+
+32x+%2B+12y+=+%28-18%29+
Thank you!
.
First we have to get the two lines in the slope intercept form,+y=mx%2Bb ("m" is the slope, "b" = y intercept)
.
First equation
.
+3x+-+8y+=+-18+
.
We will move (-8y) to the right side
.
+3x+-+8y+=+-18+ = +3x+-+8y+%2B+8y+=+-18+%2B+8y+ = +3x+=+%28-18%29+%2B+8y+
.
We will move (-18) to the left side
.
+3x+=+%28-18%29+%2B+8y+ = +3x+%2B+18+=+%28-18%29%2B18+%2B+8y+ = +3x+%2B+18+=+8y+
.
+3x+%2B+18+=+8y+ = +8y+=+3x+%2B+18++, To get it in slope intercept form, we have to divide each side by "8"
.
+8y+=+3x+%2B+18++ = +8y%2F8+=+%283x+%2B+18%29%2F8++ = +y+=+%283x+%2B+18%29%2F8++
.
+y+=+%283x+%2B+18%29%2F8++ = +y+=+%283%2F8%29x+%2B+18%2F8++ = +y+=+%283%2F8%29x+%2B+9%2F4++
.
+y+=+%283%2F8%29x+%2B+9%2F4++ is the slope intercept equation of +3x+-+8y+=+%28-18%29+, we can check our answer by replacing "x" and "y" with any point on the line, in both of the different forms of equation
.
We will use the points (-6,0)(x,y) and ( 2, 3)(x,y)( replace "x" with (-6), replace "y" with "0" for our first check, replace "x" with "2", replace "y" with "3" in our second check
.
(-6,0), +3x+-+8y+=+%28-18%29+ = +3%28-6%29+-+8%280%29+=+%28-18%29+ = +%28-18%29+-+0+=+%28-18%29+ = +%28-18%29+=+%28-18%29+ (True)
.
(-6,0), +y+=+%283%2F8%29x+%2B+9%2F4++ = +0+=+%283%2F8%29%28-6%29+%2B+9%2F4++ = +0+=+%28-18%2F8%29+%2B+9%2F4++ = +0+=+%28-9%2F4%29+%2B+9%2F4++ = +0+=+0+ (True)
.
Lets check with a different point
.
(2,3) , +3x+-+8y+=+%28-18%29+ = +3%282%29+-+8%283%29+=+%28-18%29+ = +6+-+24+=+%28-18%29+ = +%28-18%29+=+%28-18%29+(True)
.
(2,3) , +y+=+%283%2F8%29x+%2B+9%2F4++ = +3+=+%283%2F8%29%282%29+%2B+9%2F4++ = +3+=+6%2F8+%2B+9%2F4++ = +3+=+3%2F4+%2B+9%2F4++ = +3+=+12%2F4++ = +3+=+3+ (True)
.
+y+=+%283%2F8%29x+%2B+9%2F4++ is our first answer
.
Second equation (We are changing equation into slope intercept form +y+=+mx+%2B+b+)
.
+32x+%2B+12y+=+%28-18%29+
.
We need to move "32x" to the right side
.
+32x+%2B+12y+=+%28-18%29+ = +32x+-+32x+%2B+12y+=+%28-18%29+-+32x+ = +12y+=+%28-18%29+-+32x+
.
+12y+=+%28-18%29+-+32x+ = +12y+=+%28-32x%29+-+18++, To get the equation in slope intercept form, we will divide each side by "12"
.
+12y+=+%28-32x%29+-+18++ = +12y%2F12+=+%28%28-32x%29+-+18%29%2F12++ = +y+=+%28%28-32x%29+-+18%29%2F12++ = +y+=+%28-32%2F12%29x+-+18%2F12++
.
+y+=+%28-32%2F12%29x+-+18%2F12++ = +y+=+%28-8%2F3%29x+-+3%2F2++
.
The slope intercept form of the equation +32x+%2B+12y+=+%28-18%29+ is +y+=+%28-8%2F3%29x+-+3%2F2++
.
Lets check using two points again, Lets use (0, -3%2F2) and (+-9%2F16, 0)(We will do the first point first)
.
(0, -3%2F2), +32x+%2B+12y+=+%28-18%29+ = +32%280%29+%2B+12%28-3%2F2%29+=+%28-18%29+ = +0+%2B+%28-18%29+=+%28-18%29+ = +%28-18%29+=+%28-18%29+ (True)
.
(0, -3%2F2), +y+=+%28-8%2F3%29x+-+3%2F2++ = +%28-3%2F2%29+=+%28-8%2F3%29%280%29+-+3%2F2++ = +%28-3%2F2%29+=+0+-+3%2F2++ = +%28-3%2F2%29+=+%28-+3%2F2%29++ (True)
.
Second check(second point)
.
(+-9%2F16, 0) , +32x+%2B+12y+=+%28-18%29+ = +32%28-9%2F16%29+%2B+12%280%29+=+%28-18%29+ = +%28-18%29+%2B+0+=+%28-18%29+ = +%28-18%29++=+%28-18%29+ (True)
.
(+-9%2F16, 0), +y+=+%28-8%2F3%29x+-+3%2F2++ = +%280%29+=+%28-8%2F3%29%28-9%2F16%29+-+3%2F2++ = +0+=+72%2F48+-+3%2F2++ = +0+=+3%2F2+-+3%2F2++ = +0+=+0+ (True)
.
+y+=+%28-8%2F3%29x+-+3%2F2++ is our second answer.
.
Our two equations in slope intercept form are
.
+y+=+%283%2F8%29x+%2B+9%2F4++
.
+y+=+%28-8%2F3%29x+-+3%2F2++
.
The slope of the first line is +3%2F8+, the slope of the second line is +%28-8%2F3%29+
.
If two lines are parallel, their slopes would be the same(our lines are not parallel)
.
If two lines are perpendicular, their slopes are the negative reciprocal of each other
.
(examples of negative reciprocals: +%28-1%2F2%29+ and +2+, +%28-4%2F5%29+ and +5%2F4+, +%28-+4%29+ and +1%2F4+, to find the negative reciprocal of a number, switch the denominator and numerator with each other and add a negative sign)
.
Our two lines are perpendicular, +%28-8%2F3%29+ is the negative reciprocal of +3%2F8+ ( the numbers to the right of the "x" +y+=+mx+%2B+b+, Our "b's" don't have to be the same)
.
Here are the two lines in a graph( lines are in slope-intercept form)
.
+y+=+%283%2F8%29x+%2B+9%2F4++ = green line
.
+y+=+%28-8%2F3%29x+-+3%2F2++ = red line
.

.
As you can see the lines are perpendicular
.
Hope I helped, Levi