Question

sylvie finds the solution to the system of equations by graphing.
y = \\(\frac{2}{3}x + 1\\) and y = \\(-\frac{2}{3}x - 1\\)
which graph shows the solution to sylvies system of equations?
three graphs are shown, each with coordinate planes and lines, with points labeled like (-1.5, 0), (0, 1), (0, -1), (2, -4), etc. and radio buttons below them

Explanation:

Step1: Identify y-intercepts

For $y=\frac{2}{3}x+1$, y-intercept is $(0,1)$.
For $y=-\frac{2}{3}x-1$, y-intercept is $(0,-1)$.

Step2: Find intersection point

Set equations equal:
$$\frac{2}{3}x+1 = -\frac{2}{3}x-1$$
Add $\frac{2}{3}x$ to both sides:
$$\frac{4}{3}x+1 = -1$$
Subtract 1 from both sides:
$$\frac{4}{3}x = -2$$
Solve for $x$:
$$x = -2 \times \frac{3}{4} = -\frac{3}{2} = -1.5$$
Substitute $x=-1.5$ into $y=\frac{2}{3}x+1$:
$$y=\frac{2}{3}(-1.5)+1 = -1+1=0$$
Intersection: $(-1.5, 0)$

Step3: Verify slopes

$y=\frac{2}{3}x+1$ has positive slope (rises right), $y=-\frac{2}{3}x-1$ has negative slope (falls right).

Answer:

The third graph (rightmost one, with intersection $(-1.5,0)$, y-intercepts $(0,1)$ and $(0,-1)$, and correct positive/negative slopes)