Question

y = \frac{1}{3}x + 1 and y = -\frac{2}{3}x - 1\
which graph shows the solution to sylvies system of equations?

Explanation:

Step1: Identify line 1 properties

The first equation is $y=\frac{1}{3}x + 1$. It has a y-intercept at $(0,1)$ and slope $\frac{1}{3}$ (positive, so line rises right).

Step2: Identify line 2 properties

The second equation is $y=-\frac{1}{3}x - 1$. It has a y-intercept at $(0,-1)$ and slope $-\frac{1}{3}$ (negative, so line falls right).

Step3: Verify intersection point

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

Step4: Match to correct graph

Find the graph with lines passing through $(0,1)$ (rising), $(0,-1)$ (falling), and intersecting at $(-3,0)$.

Answer:

The first graph (top-left graph with points $(-1.7, 1.57)$, $(0,1)$, $(3,2)$ for the rising line; $(0,-1)$, $(2,-4)$ for the falling line, intersecting near $(-3,0)$) is the correct one.