A line that includes the points (2, g) and (-6, 7) has a slope of -7/8. What is the value of g?
Understand the Problem
The question is asking for the missing y-coordinate (g) for the point (2, g) on the line that has a given slope of -7/8 and includes the point (-6, 7). We will use the slope formula to find g.
Answer
The value of \( g \) is 0.
Answer for screen readers
The value of ( g ) is 0.
Steps to Solve
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Recall the slope formula The slope ($m$) between two points ((x_1, y_1)) and ((x_2, y_2)) is calculated using the formula: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
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Identify the points and slope From the problem, the points are ((2, g)) and ((-6, 7)), and the slope is $-\frac{7}{8}$.
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Set up the slope equation Substituting the points into the slope formula gives: $$ -\frac{7}{8} = \frac{7 - g}{-6 - 2} $$
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Simplify the equation Simplify the denominator: $$ -6 - 2 = -8 $$
So, the equation becomes: $$ -\frac{7}{8} = \frac{7 - g}{-8} $$
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Cross multiply to eliminate the fraction Cross multiplying yields: $$ -7 \cdot (-8) = 8 \cdot (7 - g) $$ This simplifies to: $$ 56 = 8(7 - g) $$
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Distribute on the right side Expanding the right side gives: $$ 56 = 56 - 8g $$
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Isolate the variable Rearranging the equation to solve for (g):
- Subtract 56 from both sides: $$ 0 = -8g $$
- Solve for g Dividing both sides by -8 results in: $$ g = 0 $$
The value of ( g ) is 0.
More Information
In this problem, we used the slope formula to find the missing y-coordinate (g) by setting up an equation based on the given slope and the coordinates of the points on the line. The slope indicates the steepness or incline of the line, which was essential in solving for ( g ).
Tips
- Misplacing values in the slope formula: Ensure each coordinate matches its corresponding variable.
- Not correctly simplifying fractions: Double-check arithmetic when cross-multiplying and isolating the variable.
- Forgetting to check your final answer: It's always a good practice to substitute back into the slope formula to ensure that the computed slope holds true with the calculated value of ( g ).
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