Photo AI
Express \( \frac{2x+1}{3} + \frac{3x-5}{2} \) as a single fraction - Junior Cycle Mathematics - Question 11 - 2015
Question 11
Express \( \frac{2x+1}{3} + \frac{3x-5}{2} \) as a single fraction. Give your answer in its simplest form. \( \frac{2x+1}{3} + \frac{3x-5}{2} = \) Solve the equat... show full transcript
Worked Solution & Example Answer:Express \( \frac{2x+1}{3} + \frac{3x-5}{2} \) as a single fraction - Junior Cycle Mathematics - Question 11 - 2015
Step 1
Express \( \frac{2x+1}{3} + \frac{3x-5}{2} \) as a single fraction.
Answer
To combine the fractions, we must find a common denominator. The least common multiple of 3 and 2 is 6.
-
Rewrite each fraction with the common denominator:
- ( \frac{2x+1}{3} = \frac{2(2x+1)}{6} = \frac{4x + 2}{6} )
- ( \frac{3x-5}{2} = \frac{3(3x-5)}{6} = \frac{9x - 15}{6} )
-
Combine the fractions: [ \frac{4x + 2 + 9x - 15}{6} = \frac{(4x + 9x) + (2 - 15)}{6} = \frac{13x - 13}{6} ]
-
Factor out the numerator: [ \frac{13(x - 1)}{6} ]
Thus, the combined fraction in simplest form is ( \frac{13(x-1)}{6} ).
Step 2
Solve the equation \( \frac{2x+1}{3} + \frac{3x-5}{2} = \frac{13}{2} \).
Answer
Using the result from part (a), we can rewrite the equation as:
[ \frac{13(x - 1)}{6} = \frac{13}{2} ]
Now, we can cross-multiply to solve for ( x ):
- Cross-multiply: [ 2(13(x - 1)) = 6(13) ]
- Expand both sides: [ 26(x - 1) = 78 ]
- Distribute and simplify: [ 26x - 26 = 78 ]
- Add 26 to both sides: [ 26x = 104 ]
- Divide by 26: [ x = 4 ]
Therefore, the solution to the equation is ( x = 4 ).