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The lengths of the sides of the triangle would be x, x, x - 5. The perimeter would be x+x+x-5 = 40, 3x = 45, x = 15. Therefore the lengths of the sides of the triangle are 15, 15, 10.

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Find a four-digit number that meets all these conditions:
a. No two digits are the same.
b. The sum of the two middle digits = the sum of the first and last digits.
c. The thousands digit is the smallest.
d. No digit is even.
e. The units digit is the largest.
1357 is an example. (This is just by trial and error.) Or 1537.

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, # liters 40% alcohol solution, and 20 - 12 = 8 liters 65% alcohol solution.

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let x = amount invested at 8%. Then the amount invested at 12% is 2x + 5000, from the given. Then
0.08x + 0.12(2x + 5000) = 4440.
0.08x+0.24x + 600 = 4440.
0.32x = 3840
x = 12,000. Thus the amount invested at 12 is $29,000.

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Substitute the bottom equation into the top equation:
3x+(-2x+4)=10.
3x-2x+4 = 10.
x = 6. Then
y = -2(6) + 4 = -12+4 = -8.

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That would be

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Let r = speed of roberto, j = speed of juana. Using the formula D = RT, (distance = rate x time), the 1st and 2nd sentences give
. The 3rd sentence implies that .
The first equation is the same as , after transposition and dividing both sides by 6. The 2nd equation is just . Adding corresponding sides of these 2 equations, we get , or r = 17.5 miles per hour. Then j=30 - r = 12.5 miles per hour.

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The equation is already written in slope-intercept form, so the slope of the line parallel to it must also be 5. Since the parallel line passes through the origin, the y-intercept must be 0. Therefore the equation of the parallel line must be y = 5x.

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The system is consistent and independent. Solving by elimination, subtract the bottom from the top equation, to get :
x = 5.
For the y-value:
4(5) - 5y = 0,
20 - 5y = 0,
5y = 20,
y = 4.

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The system is consistent and dependent, because the bottom equation is a multiple of the top equation (the top equation is multiplied by -3).
The two equations represent only one line on the cartesian plane, and thus would have all of the points on that line as solution points. There is not a single point of intersection only, but infinitely many.

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Since a regular octahedron would have 8 congruent faces, the area of each face would be 48 sq.in./8 = 6 sq.in.

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A line having undefined slope is a vertical line, so the equation is .

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Let L = the larger ineteger, S = the smaller. Then from the given,
, and .
Substituting the first equation into the second, we get .
, , and .

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Let x = ac = db. Since ac and db bisect each other at e, then e is the midpoint both of ac and db. Hence each of ae and eb will have length equal to x/2, and thus ac = db.

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let t = # hours the two cars are in motion.
Then by the formula D = RT, .
,
.
Therefore the two cars will be 150 miles apart after 10 hours.

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Let x = #number of seats in the airplane. Only seats are occupied.
Of these, are boys, are women, are girls, and 68 are men. Hence,
, or
, or
,
.
Therefore there are 240 seats in the airplane all-in-all.

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Subtract the bottom equation from the top equation, to get:
y = 3. Substitute into the top equation:
5x-3(3) = 6,
5x = 6+9,
5x = 15,
x = 3.

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Rank of column space of A = rank of A. Thus rank(A) = n (which is the number of columns of A). By the Rank-Nullity theorem, rank(A)+null(A) = n, again the number of columns of A. Therefore n + null(A) = n, and null(A) = 0.

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Consider the linear combination:
(a1)(y1)+(a2)(y2)+(a3)(y3)+....+(a(n-1))(y(n-1))+(an)(yn)= 0, a1, a2, a3,...an are scalar coefficients.
For the purpose of contradiction, suppose that {y1, y2, y3,...,yn} is a linearly dependent set. Therefore not all of a1, a2, a3, ...an are equal to zero, by definition. Since A is nonsingular, exists.
Now
(a1)(y1)+(a2)(y2)+(a3)(y3)+....+(a(n-1))(y(n-1))+(an)(yn)=
(a1)(Ax1)+(a2)(Ax2)+(a3)(Ax3)+....+(a(n-1))(Ax(n-1))+(an)(Axn)=
A((a1)(x1)+(a2)(x2)+(a3)(x3)+....+(a(n-1))(x(n-1))+(an)(xn))= 0.
Since A is nonsingular, this means that exists. Left-multiply the equation
A((a1)(x1)+(a2)(x2)+(a3)(x3)+....+(a(n-1))(x(n-1))+(an)(xn))= 0
by . This means that
(a1)(x1)+(a2)(x2)+(a3)(x3)+....+(a(n-1))(x(n-1))+(an)(xn)= 0,and
not all a1, a2, a3, ...an are equal to zero, CONTRADICTION, because {x1, x2, x3, ...xn} is a linearly independent set. Therefore
{y1, y2, y3,...,yn} has to be a linearly independent set.

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let x = speed of the river. Then 14 + x = speed of boat relative to the river DOWNSTREAM, and 14 - x = speed of the boat relative to the river upstream.
Using the formula D = RT, and knowing the fact that the times upstream and downstream are the same, then
,
,
,
,
km/hr, the speed of the river.

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40lb * $6/lb = $240
100lb * $3.32/lb = $332.
Total cost of mixture is $240 + $332 = $572.

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³√(270x)-³√
³√-³√
3³√-x³√
(3-x)³√

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let w = #students in West, e = # students in East.
Then from the given,
w = 2e -250. Also, w +e = 2858.
from the 2nd equation, w = 2858 - e. Substituting into the 1st equation,
2858 -e = 2e - 250.
3108 = 2e +e, after transposition.
3108 = 3e,
1036 = e, and w = 2858-1036 = 1822.

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let x be one number. Then the other number is 15-x.
From the given, 3x = 5(15-x)-11. Then
3x = 75 -5x-11,
8x = 64,
x = 8. The other number is 15-8 = 7.

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gallons.

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degrees.

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miles.