SOLUTION: Find values for h and k so that the system has infinitely many solutions. Write answers as integers or fractions in lowest terms. −8x − 6y = h 12x + ky = 24

Algebra ->  Finance -> SOLUTION: Find values for h and k so that the system has infinitely many solutions. Write answers as integers or fractions in lowest terms. −8x − 6y = h 12x + ky = 24       Log On


   

Question 1126889: Find values for h and k so that the system has infinitely many solutions. Write answers as integers or fractions in lowest terms.


−8x − 6y = h
12x + ky = 24



h=



k=


Found 2 solutions by MathLover1, rothauserc:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

both lines should have the same slope and y−intercept
they are one line, coincident, meaning they have all points in common
this means that there are an infinite number of solutions to the system

-8x+-6y+=+h
12x+%2B+ky+=+24
-------------------
first write both equations in slope-intercept form:
-8x+-6y+=+h
-8x+-h+=+6y
y=-%288%2F6%29x+-h%2F6+
y=-%284%2F3%29x+-h%2F6 ............eq.1=> slope is -%284%2F3%29 and y-intercept is -h%2F6

12x+%2B+ky+=+24
ky+=+24-12x+
y+=+-%2812%2Fk%29x+%2B24%2Fk.............eq.2 => slope is -%2812%2Fk%29 and y-intercept is+24%2Fk

equal slopes:
-%284%2F3%29=-%2812%2Fk%29 ...solve for k; both sides multiply by -1
4%2F3=12%2Fk........cross multiply
4k=12%2A3
4k=36
k=9
equal y-intercepts:
-h%2F6=24%2Fk+........substitute k
-h%2F6=24%2F9 ......cross multiply
-9h=24%2A6
h=144%2F-9
h=-16

so, your lines are:
y=-%284%2F3%29x+-%28-16%29%2F6 ->y=-%284%2F3%29x+%2B8%2F3
y+=+-%2812%2F9%29x+%2B24%2F9->y+=+-%284%2F3%29x+%2B8%2F3







Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
Since we have linear equations(lines), we want them to have the same slope and y intercept
:
equation 1
:
−8x − 6y = h
:
-6y = 8x +h
:
y = -8x/6 -h/6
:
y = -4x/3 -h/6
:
equation 2
:
12x + ky = 24
:
ky = -12x +24
:
y = -12x/k +24/k
:
the slopes must be equal, therefore
:
-4/3 = -12/k
:
cross multiply the fractions
:
-4k = -36
:
k = 9
:
y intercepts must be equal
:
-h/6 = 24/k
:
we know k = 9
:
-h/6 = 24/9 = 8/3
:
-3h = 48
:
h = -16
:
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h = -16 and k = 9
:
check the answer
:
equation 1
:
-8x -6y = -16
:
-4x -3y = -8
:
4x +3y = 8
:
equation 2
:
12x + 9y = 24
:
4x +3y = 8
:
answer checks
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