If \[$ Y \$\] Varies Directly With \[$ X \$\], And \[$ Y = 6x + 16 \$\] When \[$ Y = 28 \$\], What Is The Value Of \[$ X \$\]?
Understanding Direct Variation
Direct variation is a relationship between two variables, x and y, where y is a constant multiple of x. This means that as x increases or decreases, y also increases or decreases at a constant rate. The equation for direct variation is y = kx, where k is the constant of variation.
Given Equation and Value
We are given the equation y = 6x + 16 and the value of y as 28. We need to find the value of x.
Step 1: Substitute the Given Value of y into the Equation
Substitute y = 28 into the equation y = 6x + 16.
28 = 6x + 16
Step 2: Isolate the Variable x
To isolate x, we need to get rid of the constant term 16 on the right-hand side of the equation. We can do this by subtracting 16 from both sides of the equation.
28 - 16 = 6x + 16 - 16
This simplifies to:
12 = 6x
Step 3: Solve for x
Now that we have the equation 12 = 6x, we can solve for x by dividing both sides of the equation by 6.
12 / 6 = 6x / 6
This simplifies to:
2 = x
Conclusion
Therefore, the value of x is 2.
Example Use Case
Direct variation is used in many real-world applications, such as:
- Physics: The distance traveled by an object is directly proportional to the time it has been traveling.
- Economics: The cost of a product is directly proportional to the quantity produced.
- Biology: The growth rate of a population is directly proportional to the size of the population.
Tips and Tricks
- Check the units: When working with direct variation, make sure that the units of the variables are consistent.
- Use a graph: Graphing the equation y = kx can help you visualize the relationship between x and y.
- Check for extraneous solutions: When solving for x, make sure to check for extraneous solutions by plugging the value of x back into the original equation.
Common Mistakes
- Forgetting to isolate the variable: Make sure to isolate the variable x by getting rid of any constant terms on the right-hand side of the equation.
- Dividing by zero: Make sure to avoid dividing by zero when solving for x.
- Not checking for extraneous solutions: Make sure to check for extraneous solutions by plugging the value of x back into the original equation.
Conclusion
Frequently Asked Questions
Q: What is direct variation?
A: Direct variation is a relationship between two variables, x and y, where y is a constant multiple of x. This means that as x increases or decreases, y also increases or decreases at a constant rate.
Q: What is the equation for direct variation?
A: The equation for direct variation is y = kx, where k is the constant of variation.
Q: How do I determine the constant of variation (k)?
A: To determine the constant of variation (k), you can use the given values of x and y to plug into the equation y = kx. Then, solve for k.
Q: What is the difference between direct variation and inverse variation?
A: Direct variation is a relationship where y is a constant multiple of x, while inverse variation is a relationship where y is a constant divided by x.
Q: Can direct variation be represented graphically?
A: Yes, direct variation can be represented graphically as a straight line with a positive slope.
Q: How do I solve for x in a direct variation equation?
A: To solve for x, you can use the equation y = kx and plug in the given value of y. Then, solve for x.
Q: What are some real-world applications of direct variation?
A: Direct variation is used in many real-world applications, such as physics, economics, and biology.
Q: What are some common mistakes to avoid when working with direct variation?
A: Some common mistakes to avoid when working with direct variation include forgetting to isolate the variable, dividing by zero, and not checking for extraneous solutions.
Q: How do I check for extraneous solutions?
A: To check for extraneous solutions, plug the value of x back into the original equation and make sure it is true.
Q: Can direct variation be used to model real-world data?
A: Yes, direct variation can be used to model real-world data, such as the relationship between the distance traveled and the time it takes to travel.
Q: How do I determine the equation of a direct variation relationship?
A: To determine the equation of a direct variation relationship, you can use the given values of x and y to plug into the equation y = kx. Then, solve for k and write the equation in the form y = kx.
Q: What is the significance of the constant of variation (k)?
A: The constant of variation (k) represents the rate at which y changes with respect to x.
Q: Can direct variation be used to solve problems involving rates and ratios?
A: Yes, direct variation can be used to solve problems involving rates and ratios.
Q: How do I use direct variation to solve problems involving proportions?
A: To use direct variation to solve problems involving proportions, you can set up a proportion using the direct variation equation and then solve for the unknown variable.
Q: What are some examples of direct variation in real-world applications?
A: Some examples of direct variation in real-world applications include the relationship between the distance traveled and the time it takes to travel, the relationship between the cost of a product and the quantity produced, and the relationship between the growth rate of a population and the size of the population.
Q: Can direct variation be used to model complex relationships between variables?
A: Yes, direct variation can be used to model complex relationships between variables, such as the relationship between the distance traveled and the time it takes to travel, including factors such as acceleration and deceleration.
Q: How do I use direct variation to solve problems involving multiple variables?
A: To use direct variation to solve problems involving multiple variables, you can set up a system of equations using the direct variation equation and then solve for the unknown variables.
Q: What are some common pitfalls to avoid when working with direct variation?
A: Some common pitfalls to avoid when working with direct variation include forgetting to isolate the variable, dividing by zero, and not checking for extraneous solutions.
Q: Can direct variation be used to solve problems involving rates and ratios in different units?
A: Yes, direct variation can be used to solve problems involving rates and ratios in different units, such as distance and time, or cost and quantity.
Q: How do I use direct variation to solve problems involving proportions in different units?
A: To use direct variation to solve problems involving proportions in different units, you can set up a proportion using the direct variation equation and then solve for the unknown variable.
Q: What are some examples of direct variation in different fields?
A: Some examples of direct variation in different fields include the relationship between the distance traveled and the time it takes to travel in physics, the relationship between the cost of a product and the quantity produced in economics, and the relationship between the growth rate of a population and the size of the population in biology.