Question - Converting Equation to Slope-Intercept Form and Graphing
Solution:
The equation provided in the image appears to be in the form:\[ y + 6 = -\frac{3}{4}(x + 3) \]To graph this line, we need to bring it into slope-intercept form, which is $$ y = mx + b $$, where $$ m $$ is the slope and $$ b $$ is the y-intercept.We need to isolate $$ y $$ on one side of the equation:\[ y = -\frac{3}{4}(x + 3) - 6 \]To get rid of the parentheses, distribute $$ -\frac{3}{4} $$:\[ y = -\frac{3}{4}x - \frac{3}{4} \cdot 3 - 6 \]Calculating $$ -\frac{3}{4} \cdot 3 $$:\[ y = -\frac{3}{4}x - \frac{9}{4} - 6 \]We can convert 6 into fraction form to combine it with $$ -\frac{9}{4} $$:\[ -6 = -\frac{24}{4} \]So now we have:\[ y = -\frac{3}{4}x - \frac{9}{4} - \frac{24}{4} \]Combine the fractions:\[ y = -\frac{3}{4}x - \frac{33}{4} \]Now the equation is in slope-intercept form, where the slope $$ m $$ is $$ -\frac{3}{4} $$ and the y-intercept $$ b $$ is $$ -\frac{33}{4} $$.To graph this line:- Start by plotting the y-intercept at $$ 0, -\frac{33}{4} $$ (or if you prefer a decimal, approximately $$ 0, -8.25 $$).- From the y-intercept, use the slope to determine the direction and steepness of the line. The slope $$ -\frac{3}{4} $$ means that for every 4 units you move to the right (positive direction along the x-axis), you move 3 units down (negative direction along the y-axis), because the slope is negative.- Plot another point using the slope from the y-intercept and then draw the line through both points extending it across the graph.
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