1. Solve the following system by graphing. \( \begin{array}{l}x+2 y=4 \\ 3 x+y=-3\end{array} \)

To solve the system of equations by graphing: \[ \begin{cases} x + 2y = 4 \\ 3x + y = -3 \end{cases} \] we'll find the equations of both lines in slope-intercept form (**y = mx + b**) and then determine their point of intersection. ### Step 1: Rewrite Both Equations in Slope-Intercept Form **First Equation:** \[ x + 2y = 4 \] Solve for **y**: \[ 2y = -x + 4 \\ y = -\frac{1}{2}x + 2 \] **Second Equation:** \[ 3x + y = -3 \] Solve for **y**: \[ y = -3x - 3 \] ### Step 2: Graph Both Lines **First Line (\( y = -\frac{1}{2}x + 2 \)):** - **Y-intercept (0, 2):** When \( x = 0 \), \( y = 2 \). - **Slope (-1/2):** For every 2 units you move right on the x-axis, move 1 unit down on the y-axis. **Second Line (\( y = -3x - 3 \)):** - **Y-intercept (0, -3):** When \( x = 0 \), \( y = -3 \). - **Slope (-3):** For every 1 unit you move right on the x-axis, move 3 units down on the y-axis. ### Step 3: Find the Point of Intersection To find where the two lines intersect, set the equations equal to each other: \[ -\frac{1}{2}x + 2 = -3x - 3 \] **Solve for x:** \[ -\frac{1}{2}x + 3x = -3 - 2 \\ \frac{5}{2}x = -5 \\ x = -5 \times \frac{2}{5} \\ x = -2 \] **Now, solve for y using one of the equations:** \[ y = -3(-2) - 3 \\ y = 6 - 3 \\ y = 3 \] ### Step 4: Solution The two lines intersect at the point **(-2, 3)**. This is the solution to the system of equations. ### Visualization If you were to graph the lines, they would cross at the point \((-2, 3)\), confirming that this is the unique solution to the system. **Graph Representation:** ``` y | 6 | 5 | 4 | * (0,4) 3 | \ 2 | * \ 1 | \ 0 |---------*-------- x -1| \ -2| \ -3|------------* (-2,3) -4| ``` *(Note: This is a simplified text representation. For accurate graphing, using graph paper or a graphing tool/software is recommended.)*