<span>We want to check how many intersections line A and B have, that is, we want to check how many common solutions do these equations have:
</span>
i) 2x + 2y = 8
ii) x + y = 4
<span>
use equation ii) to write y in terms of x as : y=4-x,
substitute y =4-x in equation i):
</span>2x + 2y = 8
2x + 2(4-x) = 8
<span>2x+8-2x=8
8=8
this is always true, which means the equations have infinitely many common solutions.
Answer: </span><span>There are infinitely many solutions.</span><span>
</span>
Factor the equation to give
By the zero product property, either x^2=0 or (3x^2+12-6x)=0 or both.
If x^2=0, then x=0
if (3x^2+12-6x)=0, we use the quadratic formula to solve for x
where x=1 ± √ (3) i
Answer: the roots of <span>3x^4+12x^2=6x^3 are {0 (multiplicity 2), 1+√3 i, 1-√3 i}</span>
She would have to score atleast a 74 on the test for her average to be 78
1 )
1st and 2nd equation (multiply 1st by -2 ):
-2x - 10 y + 8 z = 20
2 x - y + 5 z = -9
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- 11 y + 13 z = 11 / * ( -20 )
1st and 3rd:
- 2 x - 10 y + 8 z = 20
2 x - 10 y - 5 z = 0
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- 20 x + 3 z = 20 / * 11
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220 y - 260 z = -220
- 220 y + 33 z = 220
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z = 0, y = -1, x = - 5
2 )
2 x - y + 5 = -4
- 2 x + 3 y - 5 = - 10
Substitution: y = 2 x + 9
- 2 x + 3(2 x + 9 ) = -5
4 x = - 32, x = -8, y = -7, z = 5
3 )
x + y + z = 11
1.5 x+3 y +1.5 z = 21
x = 2 y
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3 y + z = 11 / * (-2)
6 y + 1.5 z = 21
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- 6 y - 2 z = -22
6 y + 1.5 z = 21
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-0.5 z = -1, z = 2, y = 3, z = 6
Answer: the store used 6 pounds of peanuts, 3 pounds of almonds and 2 pounds of raisins.
