Understanding the Problem
Let's begin by examining the problem statement: M= -1/8, (8, -3) Find The Slope-intercept Form Of The Equation Of The Line That Has The Given Slope M. This problem provides us with two key pieces of information:
- The slope of the line (M) is -1/8.
- The line passes through the point (8, -3).
Our goal is to find the equation of this line in slope-intercept form, which is generally written as:
y = mx + b
where:
- m is the slope of the line.
- b is the y-intercept, or the value of y when x=0.
In this context, since the slope (m) is given, and a point on the line is provided, we'll determine the y-intercept (b) by substituting the known point into the slope-intercept formula.
Step-by-Step Solution
Step 1: Write the general equation
Given the slope m = -1/8, the general form of the line is:
y = -\frac{1}{8}x + b
Step 2: Substitute the known point into the equation
The point (8, -3) lies on the line, which means when x=8, y=-3. Plugging these values into the equation gives:
-3 = -\frac{1}{8} 8 + b
Step 3: Simplify the equation to solve for b
Calculate the term involving x:
-\frac{1}{8} 8 = -1
So, the equation becomes:
-3 = -1 + b
To find b, add 1 to both sides:
-3 + 1 = b
Thus:
b = -2
Final Equation in Slope-Intercept Form
Now that we have both the slope and the y-intercept, we can write the complete equation of the line:
y = -\frac{1}{8}x - 2
Additional Insights and Graphical Interpretation
Understanding the Slope
The slope -1/8 indicates that for every increase of 8 units in x, y decreases by 1 unit. The negative sign suggests the line is decreasing, slanting downward from left to right.
Understanding the Y-Intercept
The y-intercept -2 means the line crosses the y-axis at (0, -2). This point is crucial for graphing, as it provides a starting point on the y-axis from which the line extends in both directions.
Graphing the Line
Steps to Graph the Line
- Plot the y-intercept at (0, -2).
- Use the slope to find a second point: from (0, -2), move 8 units to the right (x=8) and 1 unit down (since slope is negative), reaching the point (8, -3).
- Verify that the given point (8, -3) lies on the line, confirming the correctness.
- Draw a straight line passing through these points, extending in both directions.
Alternative Methods and Tips
Using Point-Slope Form
Another approach to find the line's equation is to use the point-slope form:
y - y₁ = m(x - x₁)
Where:
- (x₁, y₁) is a point on the line, e.g., (8, -3).
- m is the slope, -1/8.
Substituting the known values:
y - (-3) = -\frac{1}{8}(x - 8)
Simplify:
y + 3 = -\frac{1}{8}x + 1
Then, subtract 3 from both sides to get the slope-intercept form:
y = -\frac{1}{8}x + 1 - 3 = -\frac{1}{8}x - 2
This confirms our earlier result, providing consistency and verifying our solution.
Common Mistakes to Avoid
- Forgetting to convert the slope fraction properly when substituting into the equation.
- Mixing up the points; ensure the coordinates are correctly identified and substituted.
- Neglecting to simplify the equation after substitution, which can lead to a less clear final answer.
- Assuming the slope-intercept form without verifying the intercept point and the slope's correctness.
Conclusion
In summary, given the slope -1/8 and a point (8, -3) on the line, we derived the equation of the line in slope-intercept form as:
y = -\frac{1}{8}x - 2
This process involved substituting the point into the slope-intercept formula, solving for the y-intercept, and confirming the consistency of the solution through alternative methods. Understanding how to find the equation of a line from a given slope and point is fundamental in coordinate geometry, and mastering these steps enhances problem-solving skills applicable across various mathematical contexts.
