Question: 6. Graph the system of linear equations to determine the values of $x$ and $y$ simultaneously satisfy both equations (if such values exist).
Note: If the solution is an ordered pair. If the solution is "no solution" or "infinitely many solutions", enter those words with spaces and spelled correctly.
y = -4x - 1
12x + 3y = -3
Solution:
1. **State the problem:** We need to find the values of $x$ and $y$ that satisfy both equations simultaneously:
$$y = -4x - 1$$
$$12x + 3y = -3$$
2. **Rewrite the second equation in slope-intercept form:**
Start with:
$$12x + 3y = -3$$
Subtract $12x$ from both sides:
$$3y = -12x - 3$$
Divide both sides by $3$:
$$y = \frac{\cancel{3}y}{\cancel{3}} = \frac{-12x - 3}{3} = -4x - 1$$
3. **Compare both equations:**
The first equation is:
$$y = -4x - 1$$
The second equation simplified is:
$$y = -4x - 1$$
Both equations represent the same line.
4. **Interpretation:**
Since both equations represent the same line, there are infinitely many solutions. Every point on the line satisfies both equations.
**Final answer:**
$$\text{infinitely many solutions}$$