1) Graph the line that passes through \( (-3,1) \) and \( (5,7) \). Find the slope of the line that passes through \( (-3,1) \) and \( (5,7) \). Label this line \( l_{1} \).

To graph the line that passes through the points \((-3, 1)\) and \((5, 7)\), and to find its slope, follow these steps: ### 1. **Calculate the Slope (\(m\))** The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in the given points \((-3, 1)\) and \((5, 7)\): \[ m = \frac{7 - 1}{5 - (-3)} = \frac{6}{8} = \frac{3}{4} \] **Slope (\(m\)) = \(\frac{3}{4}\)** ### 2. **Determine the Equation of the Line** Using the point-slope form of the equation of a line: \[ y - y_1 = m(x - x_1) \] Choose one of the points, say \((-3, 1)\), and substitute the values: \[ y - 1 = \frac{3}{4}(x - (-3)) \\ y - 1 = \frac{3}{4}(x + 3) \\ \] Simplify to get the slope-intercept form (\(y = mx + b\)): \[ y = \frac{3}{4}x + \frac{9}{4} + 1 \\ y = \frac{3}{4}x + \frac{13}{4} \] **Equation of the line (\(l_1\)):** \[ y = \frac{3}{4}x + \frac{13}{4} \] ### 3. **Graphing the Line** While I can't provide a visual graph here, I can guide you through the steps to graph it manually: 1. **Plot the Points:** - Plot the first point at \((-3, 1)\). - Plot the second point at \((5, 7)\). 2. **Draw the Line:** - Use a ruler to draw a straight line passing through both points. - This line represents \(l_1\). 3. **Label the Line:** - Label the line as \(l_1\) on your graph. ### **Summary** - **Points:** \((-3, 1)\) and \((5, 7)\) - **Slope (\(m\)):** \(\frac{3}{4}\) - **Equation of \(l_1\):** \(y = \frac{3}{4}x + \frac{13}{4}\) - **Graph:** A straight line through the points \((-3, 1)\) and \((5, 7)\), labeled \(l_1\). Feel free to use graphing paper or a graphing tool to visualize the line based on the points and the equation provided.