Certainly! Let's graph the equation [tex]\( y = \frac{3}{4}x - 6 \)[/tex] and find the intercepts and slope step-by-step.
### 1. Slope-Intercept Form
The given equation is already in the slope-intercept form, which is [tex]\( y = mx + b \)[/tex]. Here,
- [tex]\( m \)[/tex] is the slope
- [tex]\( b \)[/tex] is the y-intercept (the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex])
### 2. Finding the Slope
The slope [tex]\( m \)[/tex] is the coefficient of [tex]\( x \)[/tex]. From the equation [tex]\( y = \frac{3}{4}x - 6 \)[/tex], we can see that the slope [tex]\( m \)[/tex] is [tex]\( \frac{3}{4} \)[/tex].
### 3. Finding the Y-Intercept
The y-intercept [tex]\( b \)[/tex] is the constant term in the equation, which is -6. This means the y-intercept is at the point (0, -6).
### 4. Finding the X-Intercept
The x-intercept is the point where the graph crosses the x-axis. This happens when [tex]\( y = 0 \)[/tex].
To find the x-intercept, set [tex]\( y \)[/tex] to 0 and solve for [tex]\( x \)[/tex]:
[tex]\[ 0 = \frac{3}{4}x - 6 \][/tex]
To isolate [tex]\( x \)[/tex], add 6 to both sides:
[tex]\[ 6 = \frac{3}{4}x \][/tex]
Next, multiply both sides by the reciprocal of [tex]\( \frac{3}{4} \)[/tex], which is [tex]\( \frac{4}{3} \)[/tex]:
[tex]\[ x = 6 \times \frac{4}{3} \][/tex]
Simplifying this, we get:
[tex]\[ x = 8 \][/tex]
So, the x-intercept is at the point (8, 0).
### Conclusion
Summarizing our findings:
- Slope: [tex]\( \frac{3}{4} \)[/tex] or 0.75
- Y-Intercept: (0, -6)
- X-Intercept: (-8, 0)
With this information, you can easily graph the linear equation [tex]\( y = \frac{3}{4} x - 6 \)[/tex]. Plot the intercepts and use the slope to find another point on the line for accuracy. Then, draw a straight line through these points, and you'll have the graph of the equation.
