Solve the following two equations for $x$. Equation 13: $2\frac{1}{3} - \frac{x-2}{4} = \frac{x-1}{6}$ Equation 15: $\frac{2x-5}{5x+2} = \frac{3}{22}$

1. Problem Description

Solve the following two equations for $x$.

Equation 13: $2\frac{1}{3} - \frac{x-2}{4} = \frac{x-1}{6}$

Equation 15: $\frac{2x-5}{5x+2} = \frac{3}{22}$

2. Solution Steps

Solving Equation 13: $2\frac{1}{3} - \frac{x-2}{4} = \frac{x-1}{6}$

Step 1: Convert the mixed number to an improper fraction.

$2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$

The equation becomes:

$\frac{7}{3} - \frac{x-2}{4} = \frac{x-1}{6}$

Step 2: Find the least common multiple (LCM) of the denominators 3, 4, and

6. The LCM is

1

2.

Step 3: Multiply both sides of the equation by the LCM,

1

2. $12(\frac{7}{3} - \frac{x-2}{4}) = 12(\frac{x-1}{6})$

$12(\frac{7}{3}) - 12(\frac{x-2}{4}) = 12(\frac{x-1}{6})$

$4(7) - 3(x-2) = 2(x-1)$

$28 - 3x + 6 = 2x - 2$

Step 4: Simplify and solve for $x$.

$34 - 3x = 2x - 2$

$34 + 2 = 2x + 3x$

$36 = 5x$

$x = \frac{36}{5}$

Solving Equation 15: $\frac{2x-5}{5x+2} = \frac{3}{22}$

Step 1: Cross-multiply.

$22(2x-5) = 3(5x+2)$

Step 2: Expand both sides.

$44x - 110 = 15x + 6$

Step 3: Solve for $x$.

$44x - 15x = 6 + 110$

$29x = 116$

$x = \frac{116}{29}$

$x = 4$

3. Final Answer

For equation 13: $x = \frac{36}{5}$

For equation 15: $x = 4$