Write an equation in slope-intercept form given a point (2, -4) and a slope of -1/2.
Understand the Problem
The question is asking to derive an equation in slope-intercept form (y = mx + b) where the slope is given as -1/2 and a point (2, -4) is also provided. We will use the point-slope formula to convert it into slope-intercept form.
Answer
The equation is $y = -\frac{1}{2}x - 3$.
Answer for screen readers
The equation in slope-intercept form is:
$$ y = -\frac{1}{2}x - 3 $$
Steps to Solve
- Identify the point and slope
We are given the point ( (2, -4) ) and the slope ( m = -\frac{1}{2} ).
- Use the point-slope form
The point-slope formula is given by:
$$ y - y_1 = m(x - x_1) $$
Substituting ( m = -\frac{1}{2} ), ( x_1 = 2 ), and ( y_1 = -4 ):
$$ y - (-4) = -\frac{1}{2}(x - 2) $$
This simplifies to:
$$ y + 4 = -\frac{1}{2}(x - 2) $$
- Distribute the slope
Now, we distribute ( -\frac{1}{2} ):
$$ y + 4 = -\frac{1}{2}x + 1 $$
- Isolate ( y )
To convert to slope-intercept form, isolate ( y ) by subtracting 4 from both sides:
$$ y = -\frac{1}{2}x + 1 - 4 $$
This simplifies to:
$$ y = -\frac{1}{2}x - 3 $$
The equation in slope-intercept form is:
$$ y = -\frac{1}{2}x - 3 $$
More Information
The slope-intercept form ( y = mx + b ) allows us to easily identify the slope and the y-intercept of a linear equation. Here, the slope is ( -\frac{1}{2} ) and the y-intercept is ( -3 ).
Tips
- Forgetting to simplify: Make sure to simplify the equation fully after applying the point-slope formula.
- Incorrectly distributing the slope: Always double-check the signs when distributing the slope.
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