Solve the system of equations: { 10x + 7y = -2 { 2x - 22 = 5y

Let's solve the system of equations:

Step-by-step solution:

• Step 1: Rewrite the second equation.

Rewrite the second equation so that it is in the form \( Ax + By = C \): \[ 2x - 22 = 5y \] Subtract \( 5y \) from both sides: \[ 2x - 5y = 22 \]

• Step 2: Express one variable in terms of the other using one of the equations.

Let’s use the first equation to express \( x \) in terms of \( y \): \[ 10x + 7y = -2 \] Subtract \( 7y \) from both sides: \[ 10x = -7y - 2 \] Divide by 10: \[ x = \frac{-7y - 2}{10} \]

• Step 3: Substitute this expression into the second equation.

Substitute \( x = \frac{-7y - 2}{10} \) into \( 2x - 5y = 22 \): \[ 2\left(\frac{-7y - 2}{10}\right) - 5y = 22 \] Simplify: \[ \frac{-7y - 2}{5} - 5y = 22 \] Multiply everything by 5 to eliminate the fraction: \[ -7y - 2 - 25y = 110 \] Combine like terms: \[ -32y - 2 = 110 \] Add 2 to both sides: \[ -32y = 112 \] Divide by -32: \[ y = \frac{112}{-32} = -\frac{14}{4} = -\frac{7}{2} \] So, \( y = -\frac{7}{2} \).

• Step 4: Find the value of \( x \).

Substitute \( y = -\frac{7}{2} \) into \( x = \frac{-7y - 2}{10} \): \[ x = \frac{-7(-\frac{7}{2}) - 2}{10} \] \[ x = \frac{\frac{49}{2} - 2}{10} \] \[ x = \frac{\frac{49}{2} - \frac{4}{2}}{10} \] \[ x = \frac{\frac{45}{2}}{10} \] \[ x = \frac{45}{20} = \frac{9}{4} \] So, \( x = \frac{9}{4} \).

Answer: \( x = \frac{9}{4} \), \( y = -\frac{7}{2} \)