Converting Equation to Slope-Intercept Form and Graphing
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.
