Graphing Equations in Slope-Intercept Form
To graph the equation \( y + 6 = -\frac{3}{4}(x + 3) \), you should first rewrite it in slope-intercept form, which is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
Starting with the given equation:
\( y + 6 = -\frac{3}{4}(x + 3) \)
Isolate \( y \) on one side of the equation:
\( y = -\frac{3}{4}(x + 3) - 6 \)
Now distribute \( -\frac{3}{4} \) to both terms inside the parentheses:
\( y = -\frac{3}{4}x - \frac{3}{4} \cdot 3 - 6 \)
Multiply \( -\frac{3}{4} \) by 3 to get \( -\frac{9}{4} \), and then continue to simplify:
\( y = -\frac{3}{4}x - \frac{9}{4} - 6 \)
Since \( 6 = \frac{24}{4} \), we have:
\( y = -\frac{3}{4}x - \frac{9}{4} - \frac{24}{4} \)
Combine the constant terms:
\( y = -\frac{3}{4}x - \frac{33}{4} \)
So the equation in slope-intercept form is:
\( y = -\frac{3}{4}x - \frac{33}{4} \)
Now you can graph the line with the slope \( -\frac{3}{4} \) and a y-intercept \( -\frac{33}{4} \). Start by plotting the y-intercept on the y-axis at the point (0, -\( \frac{33}{4} \)) or (0, -8.25). Then, use the slope to find another point. From the y-intercept, move 3 units down and 4 units to the right (since the slope is negative), which will give you another point on the line. Connect these points with a straight line, and you will have graphed the given equation.
