\( \begin{array}{ll}\square \text { 1. Match each set of information to the correct equation: } & \\ \)\begin{tabular}{ll}\text { 1. parallel to }\( y=x-5 \text {, passes through }(2,2) \) & \text { a. }\( y=x-3 \) \\ \text { 2. parallel to }\( y=\frac{1}{3} x-4 \text {, passes through }(3,-1) \) & \text { b. }\( y=\frac{1}{3} x+3 \) \\ \text { 4. parallel to }\( y=x-5 \text {, passes through }(1,-2) \) & \text { c. }\( y=x \) \\ \hdashline & \text { d. }\( y=\frac{1}{3} x-2 \)\end{tabular}\end{array}

To match each set of information to the correct equation, we need to determine the equations of the lines that are parallel to the given lines and pass through the specified points. ### Step 1: Identify the slope of the given lines 1. The line \( y = x - 5 \) has a slope of \( m = 1 \). 2. The line \( y = \frac{1}{3} x - 4 \) has a slope of \( m = \frac{1}{3} \). ### Step 2: Use the point-slope form to find the equations The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope. ### Problem 1: Parallel to \( y = x - 5 \), passes through \( (2, 2) \) - Slope \( m = 1 \) - Point \( (x_1, y_1) = (2, 2) \) Using the point-slope form: \[ y - 2 = 1(x - 2) \implies y - 2 = x - 2 \implies y = x \] ### Problem 2: Parallel to \( y = \frac{1}{3} x - 4 \), passes through \( (3, -1) \) - Slope \( m = \frac{1}{3} \) - Point \( (x_1, y_1) = (3, -1) \) Using the point-slope form: \[ y + 1 = \frac{1}{3}(x - 3) \implies y + 1 = \frac{1}{3}x - 1 \implies y = \frac{1}{3}x - 2 \] ### Problem 3: Parallel to \( y = x - 5 \), passes through \( (1, -2) \) - Slope \( m = 1 \) - Point \( (x_1, y_1) = (1, -2) \) Using the point-slope form: \[ y + 2 = 1(x - 1) \implies y + 2 = x - 1 \implies y = x - 3 \] ### Step 3: Match the equations Now we can match the equations we derived with the options given: 1. **For \( y = x \)**: This matches with **c**. 2. **For \( y = \frac{1}{3}x - 2 \)**: This matches with **d**. 3. **For \( y = x - 3 \)**: This matches with **a**. ### Final Matches - 1. \( y = x \) → c - 2. \( y = \frac{1}{3}x - 2 \) → d - 3. \( y = x - 3 \) → a Thus, the matches are: - 1 → c - 2 → d - 3 → a