Sure, let's graph the line with slope [tex]\( -\frac{1}{3} \)[/tex] passing through the point [tex]\( (5, 4) \)[/tex]. Here is a step-by-step solution for how to do this:
1. Identify the Point-Slope Formula:
The point-slope formula of a line is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
2. Substitute the Given Values:
We know the slope [tex]\( m = -\frac{1}{3} \)[/tex], and the point [tex]\( (x_1, y_1) = (5, 4) \)[/tex]. Substituting these values into the equation, we get:
[tex]\[
y - 4 = -\frac{1}{3}(x - 5)
\][/tex]
3. Simplify the Equation:
Distribute [tex]\( -\frac{1}{3} \)[/tex] on the right-hand side:
[tex]\[
y - 4 = -\frac{1}{3}x + \frac{5}{3}
\][/tex]
Add 4 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -\frac{1}{3}x + \frac{5}{3} + 4
\][/tex]
Convert 4 to a fraction with a common denominator:
[tex]\[
y = -\frac{1}{3}x + \frac{5}{3} + \frac{12}{3}
\][/tex]
Combine the fractions:
[tex]\[
y = -\frac{1}{3}x + \frac{17}{3}
\][/tex]
4. Plot the Line:
- Start by plotting the point [tex]\( (5, 4) \)[/tex] on the graph.
- Use the slope to find other points. Since the slope is [tex]\( -\frac{1}{3} \)[/tex], this means that for every 3 units you move horizontally to the right, you move 1 unit vertically downwards.
Let's find another point using the slope:
- From [tex]\( (5, 4) \)[/tex], move 3 units to the right to [tex]\( x = 8 \)[/tex].
- Move 1 unit down to [tex]\( y = 3 \)[/tex].
- So, another point on the line is [tex]\( (8, 3) \)[/tex].
5. Draw the Line:
- Plot the point [tex]\( (8, 3) \)[/tex] on the graph.
- Draw a straight line passing through the two points [tex]\( (5, 4) \)[/tex] and [tex]\( (8, 3) \)[/tex].
6. Label the Graph:
- Mark the points [tex]\( (5, 4) \)[/tex] and [tex]\( (8, 3) \)[/tex] clearly.
- Write the equation of the line [tex]\( y = -\frac{1}{3}x + \frac{17}{3} \)[/tex] on the graph.
- Optionally, draw and label the x and y-axes accurately.
By following these steps, you should have a properly graph of the line passing through the point [tex]\( (5, 4) \)[/tex] with a slope of [tex]\( -\frac{1}{3} \)[/tex].
