\( \int \frac { x - 5 } { 2 } + \frac { 2 y + 1 } { 7 } + 2 = 0 \)

To solve the equation \[ \int \frac{x - 5}{2} + \frac{2y + 1}{7} + 2 = 0, \] we first need to clarify that the integral symbol \(\int\) suggests that we are integrating with respect to a variable. However, the expression inside the integral does not specify a variable of integration, and it seems to be an equation rather than an integral to evaluate. Assuming you want to solve for \(y\) in terms of \(x\), we can rewrite the equation without the integral: \[ \frac{x - 5}{2} + \frac{2y + 1}{7} + 2 = 0. \] Now, let's isolate \(y\): 1. First, simplify the equation: \[ \frac{x - 5}{2} + \frac{2y + 1}{7} + 2 = 0. \] 2. Move the other terms to the right side: \[ \frac{2y + 1}{7} = -\frac{x - 5}{2} - 2. \] 3. To combine the right side, we need a common denominator. The common denominator of 2 and 1 is 2: \[ -\frac{x - 5}{2} - 2 = -\frac{x - 5}{2} - \frac{4}{2} = -\frac{x - 5 + 4}{2} = -\frac{x - 1}{2}. \] 4. Now we have: \[ \frac{2y + 1}{7} = -\frac{x - 1}{2}. \] 5. Multiply both sides by 7 to eliminate the fraction: \[ 2y + 1 = -\frac{7(x - 1)}{2}. \] 6. Now, isolate \(2y\): \[ 2y = -\frac{7(x - 1)}{2} - 1. \] 7. To simplify further, we can express \(-1\) with a denominator of 2: \[ 2y = -\frac{7(x - 1)}{2} - \frac{2}{2} = -\frac{7(x - 1) + 2}{2}. \] 8. Finally, divide by 2 to solve for \(y\): \[ y = -\frac{7(x - 1) + 2}{4}. \] Thus, the solution for \(y\) in terms of \(x\) is: \[ y = -\frac{7(x - 1) + 2}{4}. \]