A line that includes the points (7, v) and (-3, -8) has a slope of 9/5. What is the value of v?
Understand the Problem
The question is asking for the value of v in the coordinate (7, v), given that the slope between the points (7, v) and (-3, -8) is 9/5. To solve this, we will use the slope formula: m = (y2 - y1) / (x2 - x1).
Answer
The value of $v$ is $10$.
Answer for screen readers
The value of $v$ is $10$.
Steps to Solve
-
Identify the points and slope We have two points: $(7, v)$ and $(-3, -8)$. The slope $m$ given is $\frac{9}{5}$.
-
Use the slope formula The slope formula is defined as: $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$ Here, $(x_1, y_1) = (7, v)$ and $(x_2, y_2) = (-3, -8)$.
-
Plug the points into the slope formula Substituting the values into the slope formula: $$ \frac{9}{5} = \frac{-8 - v}{-3 - 7} $$
-
Simplify the equation Calculate the denominator: $$ -3 - 7 = -10 $$ Now, the equation becomes: $$ \frac{9}{5} = \frac{-8 - v}{-10} $$
-
Cross-multiply Cross-multiplying gives: $$ 9 \cdot -10 = 5 \cdot (-8 - v) $$ This can be simplified to: $$ -90 = -40 - 5v $$
-
Isolate the variable $v$ Add $40$ to both sides: $$ -90 + 40 = -5v $$ Thus: $$ -50 = -5v $$
-
Solve for $v$ Divide both sides by $-5$: $$ v = \frac{-50}{-5} $$ Finally, we have: $$ v = 10 $$
The value of $v$ is $10$.
More Information
The slope represents the steepness of the line between two points. In this case, we found the missing coordinate by algebraically solving for $v$ using the slope formula.
Tips
- Forgetting to sign-check when simplifying can lead to errors. Always keep track of the signs throughout the process.
- Misapplying the slope formula can occur if points are mixed up; ensure the right order is maintained.
AI-generated content may contain errors. Please verify critical information
