The value of y is directly proportional to the value of x. When x = 6, y = 9. What is the value of y when x = 8?
Understand the Problem
The question asks for the value of y when x equals 8, given that y is directly proportional to x. It provides specific values for when x is 6. This requires using the concept of direct proportionality to find the unknown value.
Answer
When $x = 8$, $y = 12$.
Answer for screen readers
The value of $y$ when $x = 8$ is $12$.
Steps to Solve
- Understanding Direct Proportionality
Since $y$ is directly proportional to $x$, we can express this relationship as: $$ y = kx $$ where $k$ is the constant of proportionality.
- Find the constant of proportionality
Given that when $x = 6$, $y = 9$, we can substitute these values into the equation to find $k$: $$ 9 = k \cdot 6 $$ To find $k$, divide both sides by 6: $$ k = \frac{9}{6} = \frac{3}{2} $$
- Write the equation with the constant
Now that we have $k$, we can express the equation for any value of $x$: $$ y = \frac{3}{2} x $$
- Substitute the new value of x
To find $y$ when $x = 8$, substitute $8$ into the equation: $$ y = \frac{3}{2} \cdot 8 $$
- Calculate the value of y
Now perform the multiplication: $$ y = \frac{3 \cdot 8}{2} = \frac{24}{2} = 12 $$
The value of $y$ when $x = 8$ is $12$.
More Information
In direct proportionality, when one variable increases, the other variable increases at a constant rate. Here, the constant $\frac{3}{2}$ shows how much $y$ changes with respect to $x$.
Tips
- Confusing direct proportionality with inverse proportionality. Ensure to maintain the relationship direction.
- Forgetting to correctly calculate the constant $k$ can lead to incorrect final values.
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