Question

The equations \(y=-\frac{2}{3} x+3\) and \(2 x+3 y=9\) are equivalent. Describe how you would use each equation to graph the equation.

Short answer

The equations \(y=-\frac{2}{3} x+3\) and \(2 x+3 y=9\) are equivalent. Upon converting the second equation to slope-intercept form, you will notice they are identical. Graphically, both start from y-intercept of 3 and slope of -\frac{2}{3} (move down 2 units and right 3 units from the y-intercept) will depict the same line, hence proving they are equivalent.

Step-by-step solution

Convert the Equation in Standard Form to Slope-Intercept Form

Start with the equation in standard form, \(2x+3y=9\). To convert this to slope-intercept form, isolate y. So, subtract 2x from both sides to give \(3y = -2x+9\). Then, divide everything by 3 to provide the equation in slope-intercept form: \(y=-\frac{2}{3}x + 3\).

Identify the Slopes and Y-Intercepts from each Equation

Both equations are now in the slope-intercept form, i.e., \(y = mx + b\), where m is the slope of the line and b is the y-intercept. For the equation \(y = -\frac{2}{3}x + 3\), the slope (m) is -\frac{2}{3} and the y-intercept (b) is 3.

Graph the Equations

In a two-dimensional Cartesian system, begin plotting the line from the y-intercept (b=3). Then, from the y-intercept, move down 2 units and to the right 3 (because the slope, -\frac{2}{3}, is a negative fraction which indicates a 'fall' to the right) for each successive point on the line. The line can be created connecting these points. Notice that both of the equations produce the same line, which verifies the given statement that the two equations are indeed equivalent.