How to find the point-slope formula?
Understand the Problem
The question is asking how to derive or identify the point-slope formula, which is typically used to write the equation of a line given a point and the slope.
Answer
The point-slope formula is $y - y_1 = m(x - x_1)$.
Answer for screen readers
The point-slope formula is given by: $$ y - y_1 = m(x - x_1) $$
Steps to Solve
- Identify the components of the point-slope formula
The point-slope formula expresses the equation of a line with a slope $m$ passing through a point $(x_1, y_1)$. It can be stated as: $$ y - y_1 = m(x - x_1) $$
- Understanding the slope
The slope ($m$) represents the rate of change and is calculated by the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line.
- Applying the point-slope formula
Given a specific point $(x_1, y_1)$ and the slope $m$, plug in these values into the point-slope formula. For example, if the slope is $2$ and the point is $(3, 4)$, the equation becomes: $$ y - 4 = 2(x - 3) $$
- Rearranging to slope-intercept form (optional)
If needed, you can rearrange the equation to the slope-intercept form $y = mx + b$ for easier interpretation. For our earlier example: $$ y - 4 = 2(x - 3) $$ Distributing the slope: $$ y - 4 = 2x - 6 $$ Then, adding $4$ to both sides gives: $$ y = 2x - 2 $$
The point-slope formula is given by: $$ y - y_1 = m(x - x_1) $$
More Information
The point-slope form is particularly useful in geometry and algebra because it allows for quick construction of the linear equation given a point and a slope. It's foundational in understanding linear relationships.
Tips
- Misidentifying the components: Sometimes students confuse $x_1$ and $y_1$ values or mix up the slope. To avoid this, always clearly label your points before plugging them into the formula.
- Failing to rearrange: Students often forget to rearrange the equation into slope-intercept form when needed. Practice helps in recognizing when this rearrangement is necessary.
