\( \left. \begin{array} { l } { y q u e = 5 \perp A - y = \frac { 1 } { 2 } x - 4 } \\ { y - ( - 3 ) = - 2 ( y - 1 ) } \\ { y + 3 = - 2 x + 2 - 3 } \\ { y = 2 x - 1 } \end{array} \right. \)

To solve the system of equations given, we will analyze each equation step by step. 1. **First Equation**: \[ y = 5 \perp A - y = \frac{1}{2} x - 4 \] This equation seems to have a formatting issue. Assuming it means \( y = \frac{1}{2} x - 4 \). 2. **Second Equation**: \[ y - (-3) = -2(y - 1) \] Simplifying this: \[ y + 3 = -2y + 2 \] Rearranging gives: \[ y + 2y = 2 - 3 \implies 3y = -1 \implies y = -\frac{1}{3} \] 3. **Third Equation**: \[ y + 3 = -2x + 2 - 3 \] Simplifying this: \[ y + 3 = -2x - 1 \] Rearranging gives: \[ y = -2x - 1 - 3 \implies y = -2x - 4 \] 4. **Fourth Equation**: \[ y = 2x - 1 \] Now we have the following equations to work with: 1. \( y = \frac{1}{2} x - 4 \) 2. \( y = -\frac{1}{3} \) 3. \( y = -2x - 4 \) 4. \( y = 2x - 1 \) Next, we will substitute \( y = -\frac{1}{3} \) into the other equations to find \( x \). **Substituting into the first equation**: \[ -\frac{1}{3} = \frac{1}{2} x - 4 \] Multiplying through by 6 to eliminate the fraction: \[ -2 = 3x - 24 \implies 3x = 22 \implies x = \frac{22}{3} \] **Substituting into the third equation**: \[ -\frac{1}{3} = -2x - 4 \] Multiplying through by -1: \[ \frac{1}{3} = 2x + 4 \] Multiplying through by 3: \[ 1 = 6x + 12 \implies 6x = -11 \implies x = -\frac{11}{6} \] **Substituting into the fourth equation**: \[ -\frac{1}{3} = 2x - 1 \] Adding 1 to both sides: \[ \frac{2}{3} = 2x \implies x = \frac{1}{3} \] Now we have three different values for \( x \) corresponding to \( y = -\frac{1}{3} \): 1. \( (x, y) = \left(\frac{22}{3}, -\frac{1}{3}\right) \) 2. \( (x, y) = \left(-\frac{11}{6}, -\frac{1}{3}\right) \) 3. \( (x, y) = \left(\frac{1}{3}, -\frac{1}{3}\right) \) Thus, the solutions to the system of equations are: - \( \left(\frac{22}{3}, -\frac{1}{3}\right) \) - \( \left(-\frac{11}{6}, -\frac{1}{3}\right) \) - \( \left(\frac{1}{3}, -\frac{1}{3}\right) \) If you need further clarification or a specific solution, please let me know!