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 the variable g, given two points on a line and its slope. We need to use the formula for slope to solve for g.
Answer
$$ g = 8 $$
Answer for screen readers
$$ g = 8 $$
Steps to Solve
-
Identify points and slope We have two points: ((g, -5)) and ((4, -2)). The slope ($m$) of the line is given as $-\frac{3}{4}$.
-
Use the slope formula The formula for the slope between two points ((x_1, y_1)) and ((x_2, y_2)) is: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$
-
Substitute known values Here, ((x_1, y_1) = (g, -5)) and ((x_2, y_2) = (4, -2)). Substituting into the formula gives: $$ -\frac{3}{4} = \frac{-2 - (-5)}{4 - g} $$
-
Simplify the equation This simplifies to: $$ -\frac{3}{4} = \frac{3}{4 - g} $$
-
Cross-multiply Cross-multiply to eliminate the fraction: $$ -3(4 - g) = 12 $$
-
Distribute and solve for g Distributing gives: $$ -12 + 3g = 12 $$
Now, isolate $g$: $$ 3g = 12 + 12 $$ $$ 3g = 24 $$ $$ g = \frac{24}{3} $$ $$ g = 8 $$
$$ g = 8 $$
More Information
The value of $g$ shows the x-coordinate of the first point on the line, which aligns with the defined slope between the two points. Finding such coordinates using the slope is a common process in coordinate geometry.
Tips
- Miscalculating the slope: Ensure that the correct values are substituted corresponding to $y_2$, $y_1$, $x_2$, and $x_1$.
- Forget to consider the signs in a formula: Check that the signs are maintained properly throughout the calculations.
AI-generated content may contain errors. Please verify critical information
