word problem algebra?
Please help solve these problems and possible show step by steps.
Thanks.
1. A 12 ft rope is cut into two pieces. The lenght of the longer piece is
4 ft less than twice the lenght of the shorter piece. How long is
each piece?
2. The lenght of the first side of a triangle is 3 cm more than the lenght
of the second side. The lenght of the third side is 5cm less than
three times the lenght of the second side. The perimeter of the
triangle is 43 cm. What is the lenght of the longest side?
3. A certain triangle has two equal sides. The lenght of the third side is
8 in. less than twice the lenght of one of the two equal sides. The
perimeter of the triangle is 40 in. How long is each sides?
Comments
1) Let x represent the length of the shorter piece of rope.
The length of the longer piece of rope is 2x - 4
The total length of the rope is 12ft or the length of the longer piece plus the length of the shorter piece.
Therefore we can make the equation
(2x - 4) + x = 12
3x - 4 = 12
3x = 16
x = 5.333 ft
Longer piece = 2x - 4
= 2(5.333) - 4
= 6.666 ft
2) Let x represent the length of the second side of the triangle
The first side of the triangle is 3 + x
The third side of the triangle is 3x - 5
Therefore
(3 + x) + (x) + (3x - 5) = 43
5x - 2 = 43
5x = 45
x = 9 cm
The first side = 3 + x
= 3 + 9
= 12 cm
The third side = 3x - 5
= 3(9) - 5
= 22 cm
Therefore the length of the longest side is 22 cm.
3) Let x represent the length of each of the two equal sides
The third side is 2x - 8
Therefore
(2x - 8) + x + x = 40
4x - 8 = 40
4x = 48
x = 12 in
The third side = 2x - 8
= 2(12) - 8
= 24 - 8
= 16 in
1) Call one piece L and the other S. L + S = 12. L = 2S - 4
L = 2S - 4. Put this value of L in the first equation and solve for S.
L + S = 12.
2S - 4 + S = 12. Add 4 to both sides of the equation to get:
2S - 4 + S + 4 = 12 + 4. Do the addition and subtraction.
3S = 16. Divide both sides by 3.
S = 16/3 feet. Answer for the shorter piece.
Now put this value of S in the L + S = 12 equation and solve for L.
L + 16/3 = 12. Subtract 16/3 on both sides of the equation.
L = 12 - 16/3. 12 = 36/3.
L = 36/3 - 16/3 = 20/3. Answer for the longer piece.
2) Call side 1, X. Side 2, Y. Side 3, Z.
Perimeter of a triangle = X + Y + Z = 43.
X = Y + 3. Z = 3Y - 5. Put these values of X and Z in the perimeter equation and solve for Y.
(Y + 3) + Y + (3Y - 5) = 43. Combine the Y terms to get:
5Y - 2 = 43. Add 2 to both sides of the equation to get:
5Y = 45. Divide both sides by 5 to get:
Y = 9. Answer. Now use this value of Y in the equation X = Y + 3 and solve for X.
X = 9 + 3 = 12. Answer. Use the Y = 9 in the equation Z = 2Y - 5 and solve for Z.
Z = 2(9) - 5 = 18 - 5 = 13. Answer.
3) Call the two equal sides X. Call the third side Z.
P = 40 = X + X + Z. P = 2X + Z
Z = 2X - 8. Put this value for X in the P = 2X + Z equation.
P = 2X + (2X - 8). Put the value of P = 40 in the equation.
40 = 4X - 8. Add 8 to both sides of the equation.
48 = 4X. Divide both sides by 4.
12 = X Answer for the two equal sides. Now use X = 12 in the equation Z = 2X - 8 to find Z.
Z = 2X - 8. Z = 2(12) - 8. Z = 24 - 8 = 16. Answer for the third side.
1 is easy 12 divided by 2 length of the other piece negative 4 times L wi ch equal 4L and solve