Question

which is the graph of the equation $y - 1 = \frac{2}{3}(x - 3)$?

Explanation:

Step1: Identify the form of the equation

The given equation \( y - 1=\frac{2}{3}(x - 3) \) is in the point - slope form of a linear equation, which is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1) \) is a point on the line and \( m \) is the slope.
From the equation, we can see that the slope \( m = \frac{2}{3} \) and the line passes through the point \( (x_1,y_1)=(3,1) \).

Step2: Analyze the slope and the point

The slope \( m=\frac{2}{3} \) is positive, which means the line is increasing (as \( x \) increases, \( y \) increases). The line passes through the point \( (3,1) \).
To find another point on the line, we can use the slope. The slope \( \frac{2}{3} \) means that for a run (change in \( x \)) of 3 units, the rise (change in \( y \)) is 2 units. So, starting from the point \( (3,1) \), if we move 3 units to the right (increase \( x \) by 3, so \( x = 3+3 = 6 \)) and 2 units up (increase \( y \) by 2, so \( y=1 + 2=3 \)), we get the point \( (6,3) \).
We can also rewrite the equation in slope - intercept form (\( y=mx + b \)) to better understand the \( y \) - intercept.
Starting with \( y - 1=\frac{2}{3}(x - 3) \), we distribute the \( \frac{2}{3} \) on the right - hand side:
\( y-1=\frac{2}{3}x-2 \)
Then, we add 1 to both sides of the equation:
\( y=\frac{2}{3}x-2 + 1=\frac{2}{3}x-1 \)
So the \( y \) - intercept is \( b=- 1 \), which means the line crosses the \( y \) - axis at the point \( (0,-1) \).

To identify the graph, we look for a line that passes through the points \( (3,1) \), \( (6,3) \), and \( (0,-1) \) (or has a slope of \( \frac{2}{3} \) and passes through \( (3,1) \)) and is an increasing line (since the slope is positive).

Answer:

The graph of the equation \( y - 1=\frac{2}{3}(x - 3) \) (or \( y=\frac{2}{3}x-1 \)) is a straight line with a slope of \( \frac{2}{3} \), passing through the points \( (3,1) \), \( (0, - 1) \), \( (6,3) \) etc., and it is an increasing line. If we were to choose from a set of graphs, we would select the graph that has these characteristics (positive slope, passes through \( (3,1) \) and \( (0,-1) \)).