Write a system of equations to match each of these problems. Then solve the equations using the method... Substitution, or linear combinations.
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1. Dan is 4 years older than his brother Gabriel. Together their ages equal 22. How old is each boy.
5. The perimeter of a rectangle is 126 cm. The rectangle is twice as long as it is wide. What are the dimensions?
8. Theresa has 20 coins that are dimes and quarters. Together they equal $2.75. How many of each type of coin does she have?
15. The drama Club sold 421 tickets to its play for a total of $1,244. Student tickets cost $2 and adult tickets cost $5. How many of each type did the Drama Club sell?
17. The base of an isosceles triangle is one-third the length of each of the congruent sides. The perimeter of the triangle is 84 cm. How long is the base..
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#1
d = b+4
d+b = 22
#5
w(x+y) = 126
y = 2x
#8
d+q = 20
10d+25q = 275
#15
s+a = 421
2s+5a = 1244
#17
b = s/3
b+2s = 84
Just translate each sentence into symbols.
1. Let's define our variables. Let D be Dan's age and G be Gabriel's age. We are given that Dan is 4 years older than Gabriel, so we can write the equation D = G + 4. We are also given that their ages sum up to 22, so we can write the equation D + G = 22.
To solve this system of equations using substitution, we can substitute the expression for D from the first equation into the second equation:
(G + 4) + G = 22
2G + 4 = 22
2G = 18
G = 9
Now that we have Gabriel's age (9), we can substitute it back into the first equation to find Dan's age:
D = G + 4
D = 9 + 4
D = 13
Therefore, Gabriel is 9 years old and Dan is 13 years old.
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5. Let's call the width of the rectangle W and the length L. We are given that the rectangle is twice as long as it is wide, so L = 2W. We are also given that the perimeter of the rectangle is 126 cm, so we can write the equation 2L + 2W = 126.
Substituting L = 2W into the perimeter equation, we have:
2(2W) + 2W = 126
4W + 2W = 126
6W = 126
W = 21
Now we can substitute the value of W back into L = 2W to find the length:
L = 2W
L = 2(21)
L = 42
Therefore, the dimensions of the rectangle are 21 cm by 42 cm.
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8. Let's call the number of dimes D and the number of quarters Q. We are given that Theresa has 20 coins, so we can write the equation D + Q = 20. We are also given that the total value of the coins is $2.75, so we can write the equation 0.10D + 0.25Q = 2.75.
To solve this system of equations using linear combinations, we can multiply the first equation by -0.10 (or multiply both sides by -0.10):
-0.10(D + Q) = -0.10(20)
-0.10D - 0.10Q = -2
Now we can add this equation to the second equation:
0.10D + 0.25Q + (-0.10D) + (-0.10Q) = 2.75 + (-2)
0.15Q = 0.75
Q = 5
Now we can substitute the value of Q back into the first equation to find the number of dimes:
D + Q = 20
D + 5 = 20
D = 15
Therefore, Theresa has 15 dimes and 5 quarters.
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15. Let's call the number of student tickets S and the number of adult tickets A. We are given that the Drama Club sold 421 tickets, so we can write the equation S + A = 421. We are also given that the total revenue from ticket sales is $1,244, so we can write the equation 2S + 5A = 1244.
To solve this system of equations using linear combinations, we can multiply the first equation by -2 (or multiply both sides by -2):
-2(S + A) = -2(421)
-2S - 2A = -842
Now we can add this equation to the second equation:
2S + 5A + (-2S) + (-2A) = 1244 + (-842)
3A = 402
A = 134
Now we can substitute the value of A back into the first equation to find the number of student tickets:
S + A = 421
S + 134 = 421
S = 287
Therefore, the Drama Club sold 287 student tickets and 134 adult tickets.
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17. Let's call the length of the congruent sides L and the length of the base B. We are given that the base is one-third the length of each congruent side, so we can write the equation B = (1/3)L. We are also given that the perimeter of the triangle is 84 cm, so we can write the equation 2L + B = 84.
Substituting B = (1/3)L into the perimeter equation, we have:
2L + (1/3)L = 84
(6/3)L + (1/3)L = 84
(7/3)L = 84
L = (3/7) * 84
L = 36
Now we can substitute the value of L back into B = (1/3)L to find the length of the base:
B = (1/3)L
B = (1/3)(36)
B = 12
Therefore, the base of the isosceles triangle is 12 cm.