Question

graph the equation $y = \frac{2}{3}x$ on the coordinate plane. click to enlarge graph

Explanation:

Step1: Identify the equation type

The equation \( y = \frac{2}{3}x \) is a linear equation in slope - intercept form (\( y=mx + b \), where \( b = 0 \) and \( m=\frac{2}{3} \)). The slope \( m=\frac{2}{3} \) means for every 3 units we move to the right (increase in \( x \) by 3), we move up 2 units (increase in \( y \) by 2). Also, since \( b = 0 \), the line passes through the origin \((0,0)\).

Step2: Find two points on the line

• When \( x = 0 \), \( y=\frac{2}{3}(0)=0 \), so the point \((0,0)\) is on the line.
• When \( x = 3 \), \( y=\frac{2}{3}(3)=2 \), so the point \((3,2)\) is on the line.
• We can also find another point. When \( x=- 3 \), \( y=\frac{2}{3}(-3)=-2 \), so the point \((-3,-2)\) is on the line.

Step3: Plot the points and draw the line

Plot the points \((0,0)\), \((3,2)\) (and/or \((-3,-2)\)) on the coordinate plane. Then, draw a straight line passing through these points. The line should have a positive slope (since the slope \( \frac{2}{3} \) is positive) and pass through the origin.

Answer:

To graph \( y=\frac{2}{3}x \), plot the origin \((0,0)\) and the point \((3,2)\) (or \((-3,-2)\)) and draw a straight line through them. The line has a slope of \( \frac{2}{3} \) and passes through the origin.