1). Solve each equation.
17 = w - 4
2/3s = -6
1/3y + 1/4 = 5/12
x/8 - 1/2 = 6
5/6x - 1/3 = 5/2
2). In 1668, the Hope diamond was reduced from its original weight by about 45 carats, resulting in a diamond weighing about 67 carats. Write and solve an equation to find how many carats the original diamond weighed.
3). Write an equation to represent each relationship. The solve the equation.
Five times a number is 45.
My answers to these questions are the following:
1). 17 = w - 4, w = 19
2/3s = -6, s = 9
1/3y + 1/4 = 5/12, y = ?
x/8 - 1/2 = 6, x = ?
5/6x - 1/3 = 5/2, x = ?
2). w - 45 = 67, w = 112
The original weight of the diamond is about 112 carats.
3). 5 times (x) x = 45
2 and 3 are right.
1).
17 = w - 4, w = 19 Wrong. Add 4 to both sides of the equation.
2/3s = -6, s = 9 Wrong. Watch your sign.
1/3y + 1/4 = 5/12, y =
1/3y = 5/12 - 3/12
1/3y = 2/12
y = (2/12)/(1/3)
y = (2/12) * (3/1)
y = 6/12 = 1/2
Try the last two yourself.
x/8 - 1/2 = 6, x = ?
5/6x - 1/3 = 5/2, x = ?
1). 17 = w - 4, w = 21
2/3s = -6, s = -9
x/8 - 1/2 = 6, x = 52
5/6x - 1/3 = 5/2, x = ?
Yes, they are right.
5/6x - 1/3 = 5/2
5/6x = 5/2 + 1/3
5/6x = 15/6 + 2/6
5/6x = 17/6
x = (17/6) / (5/6)
x = (17/6) * (6/5)
x = 102/30 = 3 2/5
Check my arithmetic
To solve each equation, you need to isolate the variable on one side of the equation.
1)
a) 17 = w - 4
To isolate w, you need to get rid of the -4 on the right side of the equation. You can do this by adding 4 to both sides:
17 + 4 = w - 4 + 4
Simplifying:
21 = w
So, w = 21.
b) 2/3s = -6
To isolate s, you need to get rid of the fraction on the left side of the equation. You can do this by multiplying both sides by 3/2:
(2/3s) * (3/2) = -6 * (3/2)
Simplifying:
s = -9
So, s = -9.
c) 1/3y + 1/4 = 5/12
To isolate y, you need to get rid of the fractions. To do this, you can find a common denominator and add the fractions together. The common denominator in this case is 12:
(1/3y)(4/4) + (1/4)(3/3) = (5/12)(1)
Simplifying:
4/12y + 3/12 = 5/12
Combining like terms:
(4y + 3)/12 = 5/12
To isolate y, you can multiply both sides by 12:
12 * ((4y + 3)/12) = 12 * (5/12)
Simplifying:
4y + 3 = 5
Subtracting 3 from both sides:
4y = 2
Dividing both sides by 4:
y = 1/2
So, y = 1/2.
d) x/8 - 1/2 = 6
To isolate x, you need to get rid of the -1/2 on the left side of the equation. You can do this by adding 1/2 to both sides:
(x/8) - 1/2 + 1/2 = 6 + 1/2
Simplifying:
x/8 = 6 1/2
To convert the mixed number to an improper fraction, you can multiply the whole number (6) by the denominator (2), then add the numerator (1):
x/8 = 13/2
To isolate x, you can multiply both sides by 8:
(8)(x/8) = (8)(13/2)
Simplifying:
x = 52/2
Reducing the fraction:
x = 26
So, x = 26.
e) 5/6x - 1/3 = 5/2
To isolate x, you need to get rid of the -1/3 on the left side of the equation. You can do this by adding 1/3 to both sides:
(5/6x) - 1/3 + 1/3 = 5/2 + 1/3
Simplifying:
5/6x = 5/2 + 1/3
To find a common denominator, you can multiply the fractions:
5/6x = (5/2)(3/3) + (1/3)(2/2)
Simplifying:
5/6x = 15/6 + 2/6
Combining like terms:
5/6x = 17/6
To isolate x, you can multiply both sides by 6/5:
(6/5)(5/6x) = (6/5)(17/6)
Simplifying:
x = 17/5
So, x = 17/5.
2)
To find the original weight of the diamond, you can set up an equation. Let's say the original weight is represented by w. The weight after reducing it is 67 carats, so the equation would be:
w - 45 = 67
To solve for w, you can add 45 to both sides:
w - 45 + 45 = 67 + 45
Simplifying:
w = 112
So, the original weight of the diamond is about 112 carats.
3)
To write an equation to represent the relationship "Five times a number is 45," you need to assign a variable to the number. Let's use the variable x. The equation would be:
5x = 45
To solve for x, you can divide both sides by 5:
(5x)/5 = 45/5
Simplifying:
x = 9
So, the number is 9.