Question

a line has a slope of \\(\frac{3}{2}\\) and contains the point \\((3, -3)\\). draw the graph of this line.

Explanation:

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

The point - slope form of a line is given by \(y - y_1=m(x - x_1)\), where \(m\) is the slope and \((x_1,y_1)\) is a point on the line. Here, \(m = \frac{3}{2}\) and \((x_1,y_1)=(3,-3)\).
Substitute these values into the point - slope formula:
\(y-(-3)=\frac{3}{2}(x - 3)\)
Simplify the left - hand side: \(y + 3=\frac{3}{2}(x - 3)\)

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

Expand the right - hand side: \(y+3=\frac{3}{2}x-\frac{9}{2}\)
Subtract 3 from both sides. Since \(3=\frac{6}{2}\), we have:
\(y=\frac{3}{2}x-\frac{9}{2}-\frac{6}{2}\)
\(y=\frac{3}{2}x-\frac{15}{2}\)

Step 3: Plot the given point

The given point is \((3,-3)\). Locate \(x = 3\) on the x - axis and \(y=-3\) on the y - axis and mark the point.

Step 4: Use the slope to find another point

The slope \(m=\frac{3}{2}\) means that for a change in \(x\) (run) of \(2\) units, the change in \(y\) (rise) is \(3\) units.
Starting from the point \((3,-3)\):

• If we move \(2\) units to the right (increase \(x\) by \(2\), so \(x=3 + 2=5\)) and \(3\) units up (increase \(y\) by \(3\), so \(y=-3 + 3 = 0\)), we get the point \((5,0)\).
• If we move \(2\) units to the left (decrease \(x\) by \(2\), so \(x = 3-2 = 1\)) and \(3\) units down (decrease \(y\) by \(3\), so \(y=-3-3=-6\)), we get the point \((1,-6)\).

Step 5: Draw the line

Draw a straight line passing through the points (e.g., \((3,-3)\), \((5,0)\), \((1,-6)\)) that we have found.

(Note: Since the question asks to draw the graph, the key steps are to find the equation of the line, plot the given point, use the slope to find additional points, and then draw the line through those points. If we were to describe the graph: The line has a positive slope (\(\frac{3}{2}\)), passes through \((3,-3)\), \((5,0)\), \((1,-6)\) etc., and has a y - intercept of \(-\frac{15}{2}=-7.5\))

Answer:

To draw the line:

1. Plot the point \((3,-3)\).
2. Use the slope \(\frac{3}{2}\) (rise = 3, run = 2) to find other points (e.g., \((5,0)\) by moving 2 right and 3 up from \((3,-3)\), or \((1,-6)\) by moving 2 left and 3 down from \((3,-3)\)).
3. Draw a straight line through the plotted points. The equation of the line is \(y=\frac{3}{2}x-\frac{15}{2}\).