Question

given the slope \\(\frac{2}{3}\\) and the initial point (1,3), graph the line and choose the correct graph below. graph of a line on a coordinate plane with a grid, axes labeled from -10 to 10, and a radio button on the left side

Explanation:

Step 1: Recall the point - slope form of a line

The point - slope form of a linear equation is given by \(y - y_1=m(x - x_1)\), where \((x_1,y_1)\) is a point on the line and \(m\) is the slope of the line. Here, \(x_1 = 1\), \(y_1=3\) and \(m=\frac{2}{3}\).

Substitute these values into the point - slope formula:
\(y - 3=\frac{2}{3}(x - 1)\)

Step 2: Convert to slope - intercept form (\(y=mx + b\))

Expand the right - hand side:
\(y-3=\frac{2}{3}x-\frac{2}{3}\)

Add 3 to both sides. We know that \(3=\frac{9}{3}\), so:
\(y=\frac{2}{3}x-\frac{2}{3}+\frac{9}{3}=\frac{2}{3}x+\frac{7}{3}\approx\frac{2}{3}x + 2.33\)

Step 3: Analyze the graph

• The slope of the line is positive (\(\frac{2}{3}\)), so the line should be increasing from left to right.
• We can also use the slope to find another point on the line. Starting from the point \((1,3)\), since the slope \(m = \frac{\text{rise}}{\text{run}}=\frac{2}{3}\), we can go up 2 units and to the right 3 units. So from \((1,3)\), moving up 2 and right 3 gives us the point \((1 + 3,3+2)=(4,5)\). We can also go down 2 units and to the left 3 units. From \((1,3)\), moving down 2 and left 3 gives us the point \((1-3,3 - 2)=(-2,1)\)

Looking at the given graph, we can check if the line passes through the point \((1,3)\) and has a slope of \(\frac{2}{3}\). Let's check the y - intercept. From the equation \(y=\frac{2}{3}x+\frac{7}{3}\), when \(x = 0\), \(y=\frac{7}{3}\approx2.33\), which matches the y - intercept of the given graph. Also, the slope of the line in the graph is positive and the line passes through the point \((1,3)\) (we can verify by substituting \(x = 1\) into \(y=\frac{2}{3}x+\frac{7}{3}\): \(y=\frac{2}{3}(1)+\frac{7}{3}=\frac{2 + 7}{3}=3\))

So the given graph (the one with the line passing through the point that can be derived from the slope - point form and has the correct slope and y - intercept) is the correct graph.

Answer:

The graph (the one with the line that has a positive slope, passes through \((1,3)\) and has a y - intercept of \(\frac{7}{3}\)) is the correct graph. (Since the graph is shown with a line that has a slope of \(\frac{2}{3}\) and passes through \((1,3)\), the correct graph is the one provided in the image with the line drawn as described)