Which is the graph of the equation y−1=2/3(x−3)?
The Correct Answer and Explanation is:
To graph the equation ( y – 1 = \frac{2}{3}(x – 3) ), we start by rewriting it in the slope-intercept form, ( y = mx + b ). Here’s how we can do that step by step:
- Distributing the Slope: We start with the equation in point-slope form:
[
y – 1 = \frac{2}{3}(x – 3)
]
Next, we distribute ( \frac{2}{3} ) on the right side:
[
y – 1 = \frac{2}{3}x – 2
] - Isolating ( y ): To isolate ( y ), we add ( 1 ) to both sides:
[
y = \frac{2}{3}x – 2 + 1
]
Simplifying this gives:
[
y = \frac{2}{3}x – 1
]
Now, we can see that the slope ( m ) is ( \frac{2}{3} ) and the y-intercept ( b ) is ( -1 ). - Plotting the Graph: With the slope and y-intercept, we can plot the graph:
- Start at the y-intercept ((0, -1)).
- From this point, use the slope ( \frac{2}{3} ) to find another point. This means that for every 3 units you move to the right (positive direction along the x-axis), you move up 2 units (positive direction along the y-axis).
- From ((0, -1)), moving 3 units to the right takes you to ( (3, -1) ), and then moving up 2 units brings you to ( (3, 1) ).
- Finalizing the Line: Draw a line through the points ( (0, -1) ) and ( (3, 1) ). This line extends infinitely in both directions.
- Additional Points: For further accuracy, you could find additional points by substituting other x-values into the equation.
Thus, the graph of the equation ( y – 1 = \frac{2}{3}(x – 3) ) is a straight line with a slope of ( \frac{2}{3} ) and a y-intercept at (-1).