Question - Graphing a Line in Slope-Intercept Form
Solution:
To graph the line given by the equation:\[ y + 3 = \frac{9}{4}(x + 4) \]We should first rewrite it into the slope-intercept form, which is $$ y = mx + b $$, where $$ m $$ is the slope and $$ b $$ is the y-intercept.Let's solve the equation for $$ y $$:\[ y = \frac{9}{4}(x + 4) - 3 \]Now, distribute the $$ \frac{9}{4} $$ across $$ (x + 4) $$:\[ y = \frac{9}{4}x + \frac{9}{4} \cdot 4 - 3 \]Simplify the constants:\[ y = \frac{9}{4}x + 9 - 3 \]\[ y = \frac{9}{4}x + 6 \]Now we have the equation in slope-intercept form with a slope $$ m = \frac{9}{4} $$ and a y-intercept $$ b = 6 $$.To graph this line, follow these steps:1. Start by plotting the y-intercept (0,6) on the graph.2. From this point, use the slope to determine the next point. The slope is $$ \frac{9}{4} $$ which means that for every 4 units you move to the right (in the positive x-direction), you move 9 units up (in the positive y-direction).3. Plot another point using the slope. For example, starting at (0,6), go right 4 units to (4,6) and then up 9 units to (4,15).4. Draw a straight line through the points to complete the graph of the line. Remember, you can plot more points if needed to ensure accuracy before drawing your line.
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