Solve for x in each of the following:
a: 2/3x = 11/6
b: 3/4 / x = 1/3
c: 5/6 - 2/3x = 3/4
d: 2x/3 + 1/4 = x/6 - 1/2
Please help me and explain, I don't get fractions at all!
for each one, multiply each term by the LCM
e.g.
c:
5/6 - 2/3 x = 3/4
LCM = 12
so multiply each term by 12
(5/6)(12) - (2x/3)(12) = (3/4)(12)
10 - 8x = 9
-8x = -1
x = 1/8
do the others the same way
in b) I would cross-multply first to get
(1/3)x = 3/4 , the proceed by multiplying by 12
I am mainly confused with problem D. I don't understand and have yet to get the right answer.
a: To solve for x in the equation 2/3x = 11/6, you can follow these steps:
Step 1: Get rid of the fraction by multiplying both sides of the equation by the reciprocal of 2/3, which is 3/2:
(3/2)(2/3x) = (3/2)(11/6)
On the left side, the 2/3 cancels out with the 3/2, leaving just x:
x = (3/2)(11/6)
Step 2: Simplify the expression on the right side by multiplying the numerators and the denominators:
x = (3 * 11) / (2 * 6)
Step 3: Evaluate the expression on the right side:
x = 33 / 12
Step 4: Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
x = (33 ÷ 3) / (12 ÷ 3)
x = 11 / 4
So the solution to the equation is x = 11/4.
b: To solve for x in the equation (3/4) / x = 1/3, you can use the concept of reciprocals:
Step 1: Multiply both sides of the equation by the reciprocal of (3/4), which is (4/3):
(4/3) * ((3/4) / x) = (4/3) * (1/3)
On the left side, the (3/4) and (4/3) cancel out, leaving just 1/x:
1/x = (4/3) * (1/3)
Step 2: Simplify the expression on the right side by multiplying the numerators and the denominators:
1/x = 4/9
Step 3: To get x by itself, take the reciprocal of both sides of the equation:
1 / (1/x) = 1 / (4/9)
On the left side, the reciprocal of 1/x is just x:
x = 1 / (4/9)
Step 4: Invert and multiply to divide by a fraction:
x = 1 * (9/4)
Step 5: Simplify the expression on the right side by multiplying the numerators and the denominators:
x = 9/4
So the solution to the equation is x = 9/4.
c: To solve for x in the equation 5/6 - (2/3)x = 3/4, you can follow these steps:
Step 1: Simplify both sides of the equation by finding a common denominator. The least common multiple of 6 and 4 is 12.
(5/6) - (2/3)x = (3/4)
Multiply the numerators and denominators of (5/6) by 2 to get a common denominator of 12:
(10/12) - (2/3)x = (3/4)
Step 2: Simplify the equation further by multiplying the numerators and denominators of (3/4) by 3:
(10/12) - (2/3)x = (9/12)
Step 3: To eliminate the denominators, you can multiply every term in the equation by the common denominator, which is 12:
12 * [(10/12) - (2/3)x] = 12 * (9/12)
On the left side, the 12s cancel out:
10 - 8x = 9
Step 4: Isolate the variable term by subtracting 10 from both sides of the equation:
10 - 10 - 8x = 9 - 10
-8x = -1
Step 5: Solve for x by dividing both sides of the equation by -8:
x = (-1) / (-8)
Simplifying the expression on the right side gives:
x = 1/8
So the solution to the equation is x = 1/8.
d: To solve for x in the equation (2x/3) + (1/4) = (x/6) - (1/2), follow these steps:
Step 1: Start by getting rid of the fractions by finding a common denominator for 3, 4, and 6. The least common multiple is 12.
Multiply every term in the equation by 12 to eliminate the denominators:
12 * [(2x/3) + (1/4)] = 12 * [(x/6) - (1/2)]
This simplifies to:
4x + 3 = 2x - 6
Step 2: Simplify the equation by combining like terms. Subtract 2x from both sides and add 6 to both sides:
4x - 2x + 3 + 6 = 2x - 2x - 6 + 6
This simplifies to:
2x + 9 = 0
Step 3: Isolate the variable term by subtracting 9 from both sides of the equation:
2x + 9 - 9 = 0 - 9
This simplifies to:
2x = -9
Step 4: Solve for x by dividing both sides of the equation by 2:
2x / 2 = -9 / 2
This simplifies to:
x = -9 / 2
So the solution to the equation is x = -9 / 2.