1. **Stating the problem:** Solve the system of linear equations: $$\begin{cases} 3x + y = 6 \\ x + y = 4 \\ 3x - y = 7 \end{cases}$$ 2. **Formula and rules:** To solve a system of linear equations, we can use substitution or elimination methods. Here, we will use elimination. 3. **Step 1: Use the first two equations:** $$3x + y = 6 \quad (1)$$ $$x + y = 4 \quad (2)$$ Subtract equation (2) from equation (1) to eliminate $y$: $$ (3x + y) - (x + y) = 6 - 4 $$ $$ 3x + y - x - y = 2 $$ $$ 2x = 2 $$ $$ x = 1 $$ 4. **Step 2: Substitute $x=1$ into equation (2):** $$ 1 + y = 4 $$ $$ y = 3 $$ 5. **Step 3: Verify with the third equation:** $$ 3x - y = 7 $$ Substitute $x=1$, $y=3$: $$ 3(1) - 3 = 3 - 3 = 0 \neq 7 $$ This means the three equations are inconsistent and do not have a common solution. 6. **Interpretation:** The first two equations intersect at $(1,3)$, but the third equation represents a line that does not pass through this point. **Final answer:** The system of three equations is inconsistent; there is no single solution that satisfies all three simultaneously. --- **Additional info about the pink line $3x + y = 6$:** - Passes through points $(0,6)$, $(1,3)$, and $(2,0)$. - X-axis range: $-3$ to $7$. - Y-axis range: $-1$ to $7$. - Position hints for points: top-left for $(0,6)$, center for $(1,3)$, bottom-right for $(2,0)$.