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Math  /  Algebra

Question3t−24−2t+33=23−t\frac{3 t-2}{4}-\frac{2 t+3}{3}=\frac{2}{3}-t

Studdy Solution

STEP 1

1. The equation 3t−24−2t+33=23−t\frac{3t-2}{4} - \frac{2t+3}{3} = \frac{2}{3} - t involves rational expressions.
2. We need to find the value of t t that satisfies the equation.
3. The equation requires finding a common denominator to combine terms effectively.

STEP 2

1. Eliminate fractions by finding a common denominator.
2. Simplify and combine like terms.
3. Solve for the variable t t .
4. Check the solution by substituting it back into the original equation.

STEP 3

Identify the least common denominator (LCD) for the fractions involved in the equation. The denominators are 4, 3, and 1 (for the term −t-t), so the LCD is 12.
Multiply every term in the equation by 12 to eliminate the fractions:
12(3t−24)−12(2t+33)=12(23)−12(t) 12 \left(\frac{3t-2}{4}\right) - 12 \left(\frac{2t+3}{3}\right) = 12 \left(\frac{2}{3}\right) - 12(t)

STEP 4

Simplify each term:
3(3t−2)−4(2t+3)=4(2)−12t 3(3t - 2) - 4(2t + 3) = 4(2) - 12t

STEP 5

Distribute the constants in each term:
9t−6−8t−12=8−12t 9t - 6 - 8t - 12 = 8 - 12t

STEP 6

Combine like terms on the left side:
(9t−8t)−6−12=8−12t (9t - 8t) - 6 - 12 = 8 - 12t
t−18=8−12t t - 18 = 8 - 12t

STEP 7

Add 12t 12t to both sides to get all terms involving t t on one side:
t+12t−18=8 t + 12t - 18 = 8
13t−18=8 13t - 18 = 8

STEP 8

Add 18 to both sides to isolate the term with t t :
13t=26 13t = 26

STEP 9

Divide both sides by 13 to solve for t t :
t=2 t = 2

STEP 10

Check the solution by substituting t=2 t = 2 back into the original equation:
Original equation: 3t−24−2t+33=23−t\frac{3t-2}{4} - \frac{2t+3}{3} = \frac{2}{3} - t
Substitute t=2 t = 2 :
3(2)−24−2(2)+33=23−2 \frac{3(2)-2}{4} - \frac{2(2)+3}{3} = \frac{2}{3} - 2
Simplify each term:
6−24−4+33=23−2 \frac{6-2}{4} - \frac{4+3}{3} = \frac{2}{3} - 2
44−73=23−2 \frac{4}{4} - \frac{7}{3} = \frac{2}{3} - 2
1−73=23−63 1 - \frac{7}{3} = \frac{2}{3} - \frac{6}{3}
1−73=−43 1 - \frac{7}{3} = -\frac{4}{3}
Convert 1 to 33\frac{3}{3}:
33−73=−43 \frac{3}{3} - \frac{7}{3} = -\frac{4}{3}
−43=−43 -\frac{4}{3} = -\frac{4}{3}
Both sides are equal, confirming the solution is correct.
The solution is:
2 \boxed{2}

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