Problem 61 - Precalculus Essentials

Question

a. Rewrite the given equation in slope-intercept form. b. Give the slope and y-intercept. c. Use the slope and y-intercept to graph the linear function. $$2 x+3 y-18=0$$

Step-by-Step Solution

Answer

The slope-intercept form of the given equation is \(y = -\frac{2}{3}x + 6\). The slope is -\frac{2}{3} and the y-intercept is 6. The line crosses the y-axis at (0,6) and then every step 3 unit right it moves 2 units down.

Step 1: Rewrite Equation in Slope-Intercept Form

This requires us to rearrange the given equation \(2x + 3y - 18 = 0\) in the form \(y = mx + c\). This is done by isolating y. \nFirst, subtract \(2x\) from both sides to get: \n\[3y = -2x + 18\]\nThen divide by 3 to get: \n\[y = -\frac{2}{3}x + 6\]

Step 2: Identify the Slope and Y-Intercept

From the obtained equation \(y = -\frac{2}{3}x + 6\), \nSlope (m) = -\frac{2}{3} and Y-intercept (c) = 6.

Step 3: Graph the Linear Function

Begin graphing by labeling the y-intercept, which is the point (0,6) on the y-axis. This is where the line crosses the y-axis. From this point, move down 2 units (because of the negative slope) and then right 3 units (representing the denominator of the slope fraction). Mark that point. Draw a line through the two points, which represents the graph of the linear function.