The points (g, -5) and (4, -2) fall on a line with a slope of -3/4. What is the value of g?
Understand the Problem
The question is asking for the value of g in the point (g, -5) such that the slope between this point and the point (4, -2) equals -3/4. To solve this, we can use the slope formula, which is (y2 - y1) / (x2 - x1).
Answer
The value of \(g\) is \(8\).
Answer for screen readers
The value of (g) is (8).
Steps to Solve
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Identify the points and slope We have two points: ((g, -5)) and ((4, -2)). The slope (m) is given as (-\frac{3}{4}).
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Use the slope formula The slope formula is given by:
$$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
Here, let ((x_1, y_1) = (g, -5)) and ((x_2, y_2) = (4, -2)). Substitute the values into the slope formula:
$$ -\frac{3}{4} = \frac{-2 - (-5)}{4 - g} $$
- Simplify the right side of the equation Calculating the numerator:
$$ -2 - (-5) = -2 + 5 = 3 $$
So the equation now becomes:
$$ -\frac{3}{4} = \frac{3}{4 - g} $$
- Cross-multiply to solve for (g) Cross-multiplying gives:
$$ -3(4 - g) = 4 \times 3 $$
This simplifies to:
$$ -3(4 - g) = 12 $$
- Distribute and simplify Distributing the (-3) leads to:
$$ -12 + 3g = 12 $$
- Solve for (g) Add 12 to both sides:
$$ 3g = 24 $$
Now, divide by 3:
$$ g = 8 $$
The value of (g) is (8).
More Information
This solution uses the definition of slope and properties of basic algebra. Finding a missing coordinate with a provided slope is a common type of problem in algebra.
Tips
- Miscalculating the numerator or denominator in the slope formula.
- Forgetting to distribute negative signs when simplifying.
- Not cross-multiplying correctly, which can lead to wrong equations.
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