System Graph Check 42E75C

1. **State the problem:** We need to determine which of the given systems of equations could be represented by a graph consisting of one descending straight line and one vertical line intersecting at one point. 2. **Recall the characteristics of lines:** - A vertical line has an equation of the form $x = c$, where $c$ is a constant. - A descending line has a negative slope. 3. **Analyze each system:** **System 1:** $$\begin{cases} x + 2y = 8 \\ x = -5 \end{cases}$$ - The second equation $x = -5$ is a vertical line. - The first equation can be rewritten as $2y = 8 - x$ or $y = \frac{8 - x}{2}$. - This is a line with slope $-\frac{1}{2}$ (since $y = -\frac{1}{2}x + 4$), which is descending. - So system 1 has one vertical and one descending line. **System 2:** $$\begin{cases} y = -7x + 13 \\ y = -1 \end{cases}$$ - Both equations are in terms of $y$. - The first is a descending line with slope $-7$. - The second is a horizontal line $y = -1$. - No vertical line here, so this system does not match the graph. **System 3:** $$\begin{cases} 3x = 8 \\ 3x + y = 15 \end{cases}$$ - The first equation is $x = \frac{8}{3}$, a vertical line. - The second can be rewritten as $y = 15 - 3x$. - This is a line with slope $-3$, which is descending. - So system 3 matches the graph. **System 4:** $$\begin{cases} y = 2x - 7 \\ 4 + y = 12 \end{cases}$$ - The first equation has slope $2$, which is ascending. - The second simplifies to $y = 8$, a horizontal line. - No vertical line here, so this system does not match the graph. 4. **Conclusion:** - Systems 1 and 3 could be represented by the described graph. - Systems 2 and 4 could not. 5. **Final answer:** - Yes: Systems 1 and 3 - No: Systems 2 and 4