Question: Graph the system of linear equations to determine the values of $x$ and $y$ that simultaneously satisfy both equations (if such values exist).
Note: If the solution is an ordered pair, do not put spaces between the comma and the second value. If the solution is "no solution" or "infinitely many solutions," enter those words with spaces and spelled correctly.
$y + 4 = 3x$
$y = - \frac{1}{2} x + 3$
1. **State the problem:**
We need to find the values of $x$ and $y$ that satisfy both equations simultaneously:
$$y + 4 = 3x$$
$$y = - \frac{1}{2} x + 3$$
2. **Rewrite the first equation in slope-intercept form ($y = mx + b$):**
$$y + 4 = 3x$$
Subtract 4 from both sides:
$$y = 3x - 4$$
3. **Now we have the system:**
$$y = 3x - 4$$
$$y = - \frac{1}{2} x + 3$$
4. **Set the right-hand sides equal to find $x$: since both equal $y$,**
$$3x - 4 = - \frac{1}{2} x + 3$$
5. **Solve for $x$: add $\frac{1}{2} x$ to both sides:**
$$3x + \frac{1}{2} x - 4 = 3$$
Combine like terms:
$$\left(3 + \frac{1}{2}\right) x - 4 = 3$$
$$\frac{6}{2} x + \frac{1}{2} x - 4 = 3$$
$$\frac{7}{2} x - 4 = 3$$
6. **Add 4 to both sides:**
$$\frac{7}{2} x = 7$$
7. **Divide both sides by $\frac{7}{2}$:**
$$x = \frac{7}{\frac{7}{2}} = 7 \times \frac{2}{7} = 2$$
8. **Substitute $x=2$ into one of the original equations to find $y$:**
Using $y = 3x - 4$:
$$y = 3(2) - 4 = 6 - 4 = 2$$
9. **Solution:**
The values that satisfy both equations simultaneously are:
$$(x,y) = (2,2)$$