A line with a slope of -1 passes through the points (w, -7) and (-3, -9). What is the value of w?
Understand the Problem
The question asks us to determine the missing coordinate 'w' of a point on a line that has a specific slope. Given the slope of -1 and another point, we can apply the slope formula to find the missing value.
Answer
\( w = -5 \)
Answer for screen readers
The value of ( w ) is ( -5 ).
Steps to Solve
-
Identify the Points and Slope We have two points: $(w, -7)$ and $(-3, -9)$. The slope ($m$) is given as $-1$.
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Use the Slope Formula The slope formula is given by:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
We can substitute the values:
- $m = -1$
- $y_1 = -7$, $y_2 = -9$
- $x_1 = w$, $x_2 = -3$
Substituting these into the formula gives:
$$ -1 = \frac{-9 - (-7)}{-3 - w} $$
- Simplify the Equation Now, simplify the numerator:
$$ -1 = \frac{-9 + 7}{-3 - w} $$
This simplifies to:
$$ -1 = \frac{-2}{-3 - w} $$
- Cross-Multiply Cross-multiplying to remove the fraction gives:
$$ -1 \cdot (-3 - w) = -2 $$
- Distribute and Solve for w Distributing the left side leads to:
$$ 3 + w = -2 $$
Now, isolate $w$:
$$ w = -2 - 3 $$
So,
$$ w = -5 $$
The value of ( w ) is ( -5 ).
More Information
The slope of a line indicates the steepness and direction. In this case, a slope of -1 indicates that for every unit increase in x, y decreases by 1 unit. The points were solved using the slope formula, which is fundamental in coordinate geometry.
Tips
Common mistakes include:
- Forgetting to apply the correct signs when subtracting coordinates.
- Misplacing coordinates in the slope formula. Always double-check the order of coordinates used.
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