Unil 7 Test Review. Solving Systems of Linear Equations and inequalifies 1. Delermine how mony solulions exis for sach of ihe bystems of linear equations, if onty one whation, list if. a. \( \left\{\begin{array}{l}y=3 x=4 \\ y=-\frac{1}{2} x+3\end{array}\right. \) b. \( \left\{\begin{array}{l}2 x+y=2 \\ 2 x+y=-2\end{array}\right. \) e. \( \left\{\begin{array}{l}x+3 y=3 \\ y=x=1\end{array}\right. \) d. \( \left\{\begin{array}{l}y=x+4 \\ y-4=x\end{array}\right. \)

Sure, let's analyze each system of linear equations to determine the number of solutions they have. If a system has exactly one solution, we'll also find that solution. --- ### **Problem 1: Determine the number of solutions for each system. If there is only one solution, list it.** #### **a.** \[ \begin{cases} y = 3x + 4 \\ y = -\frac{1}{2}x + 3 \end{cases} \] **Solution:** To find the point of intersection, set the two equations equal to each other: \[ 3x + 4 = -\frac{1}{2}x + 3 \] **Solving for \( x \):** \[ 3x + \frac{1}{2}x = 3 - 4 \] \[ \frac{7}{2}x = -1 \] \[ x = -\frac{2}{7} \] **Substitute \( x \) back into one of the equations to find \( y \):** Using \( y = 3x + 4 \): \[ y = 3\left(-\frac{2}{7}\right) + 4 = -\frac{6}{7} + \frac{28}{7} = \frac{22}{7} \] **Conclusion:** - **Number of solutions:** Exactly one. - **Solution:** \( \left( -\frac{2}{7}, \frac{22}{7} \right) \) --- #### **b.** \[ \begin{cases} 2x + y = 2 \\ 2x + y = -2 \end{cases} \] **Solution:** Subtract the second equation from the first: \[ (2x + y) - (2x + y) = 2 - (-2) \] \[ 0 = 4 \] This is a contradiction, meaning there's no solution. **Conclusion:** - **Number of solutions:** No solution (inconsistent system). --- #### **c.** \[ \begin{cases} x + 3y = 3 \\ y = x + 1 \end{cases} \] **Solution:** Substitute \( y = x + 1 \) into the first equation: \[ x + 3(x + 1) = 3 \] \[ x + 3x + 3 = 3 \] \[ 4x + 3 = 3 \] \[ 4x = 0 \] \[ x = 0 \] Now, find \( y \): \[ y = 0 + 1 = 1 \] **Conclusion:** - **Number of solutions:** Exactly one. - **Solution:** \( (0, 1) \) --- #### **d.** \[ \begin{cases} y = x + 4 \\ y - 4 = x \end{cases} \] **Solution:** Simplify the second equation: \[ y - 4 = x \Rightarrow y = x + 4 \] Both equations are identical: \[ y = x + 4 \] **Conclusion:** - **Number of solutions:** Infinitely many solutions (the equations represent the same line). --- ### **Summary:** - **a.** One solution: \( \left( -\frac{2}{7}, \frac{22}{7} \right) \) - **b.** No solution - **c.** One solution: \( (0, 1) \) - **d.** Infinitely many solutions If you have any further questions or need more explanations, feel free to ask!