Fractional equations questions with answers

Questions

1. Question 1One fraction

   Solve $\dfrac{x - 3}{5} = -5$.

   Show the worked solution

   1. Step 1: Multiply both sides by the denominator$x - 3 = -25$
   2. Step 2: Move the number across (it changes sign)$x = -22$
2. Question 2A fraction on each side

   Solve $\dfrac{4x - 6}{6} = \dfrac{x - 5}{5}$.

   Show the worked solution

   1. Step 1: Cross-multiply: each top times the other side's bottom$5(4x - 6) = 6(x - 5)$
   2. Step 2: Expand both sides$20x - 30 = 6x - 30$
   3. Step 3: Collect the x terms on the left and the numbers on the right (a term changes sign as it crosses)$20x - 6x = -30 + 30$
   4. Step 4: Simplify both sides$14x = 0$
   5. Step 5: Divide both sides by the number in front of x$x = 0$
3. Question 3Two fractions in x

   Solve $\dfrac{x}{6} + \dfrac{x}{2} = 2$.

   Show the worked solution

   1. Step 1: Find the LCM of the denominators$\text{LCM of } 6 \text{ and } 2 =$ $6$
   2. Step 2: Multiply every term by the LCM$x + 3x = 12$
   3. Step 3: Collect the x terms$4x = 12$
   4. Step 4: Divide both sides by the number in front of x$x = 3$
4. Question 4x in the denominator

   Solve $\dfrac{7}{x - 3} = \dfrac{9}{x + 1}$.

   Show the worked solution

   1. Step 1: Cross-multiply: each top times the other side's bottom$7(x + 1) = 9(x - 3)$
   2. Step 2: Expand both sides$7x + 7 = 9x - 27$
   3. Step 3: Collect the x terms on the left and the numbers on the right (a term changes sign as it crosses)$7x - 9x = -27 - 7$
   4. Step 4: Simplify both sides$-2x = -34$
   5. Step 5: Divide both sides by the number in front of x$x = 17$
5. Question 5The solution is given: find the constant k

   Given that $x = 3$ is the solution of the equation $\dfrac{4x + k - 8}{4} = \dfrac{4x - 9}{6}$, find the value of the constant $k$.

   Show the worked solution

   1. Step 1: Cross-multiply: each top times the other side's bottom$6(4x + k - 8) = 4(4x - 9)$
   2. Step 2: Expand both sides$24x + 6k - 48 = 16x - 36$
   3. Step 3: x is the solution: put $x = 3$ into both sides and simplify$6k + 24 = 12$
   4. Step 4: Collect the k terms on the left and the numbers on the right (a term changes sign as it crosses)$6k = 12 - 24$
   5. Step 5: Simplify both sides$6k = -12$
   6. Step 6: Divide both sides by the number in front of k$k = -2$