Question

It is known that $y$ is proportional to $x$. Experimental measurements are recorded in Table $7,2.$ $$\text{Unknown environment 'tabular'}$$ (a) Determine the equation connecting $y$ and $x$. (b) Calculate $y$ when $x=2$.

Short answer

Answer: The equation connecting y and x is y = 6x, and the value of y when x=2 is y = 12.

Step-by-step solution

Understanding the relationship between y and x

As given, y is proportional to x. This type of relationship can be expressed as y=kx, where k is the constant of proportionality.

Determine the constant of proportionality, k

We can find the constant of proportionality, k, using the data given in the table. Let's pick the first data point, where y=30 and x=5. Using the equation y=kx, we have: 30 = k(5) Now, isolate k by dividing both sides by 5: k = 30/5 k = 6

Write the equation connecting y and x

Now that we have found the value of the constant of proportionality, k, we can write the equation connecting y and x as follows: y = 6x

Calculate y when x=2

Using the equation y=6x, we will now calculate the value of y when x=2: y = 6(2) y = 12 So when x=2, the value of y is 12. The equation connecting y and x is y = 6x, and the value of y when x=2 is y = 12.

Key concepts

Constant of Proportionality

A constant of proportionality is a key concept in understanding proportional relationships between two variables, such as in the equation $y=kx$. This equation means that the variable $y$ changes at a constant rate $k$ as $x$ changes.
When $y$ is proportional to $x$, it implies that doubling $x$ would result in doubling $y$, tripling $x$ would triple $y$, and so on. The rate $k$, or the constant of proportionality, remains unchanged in all such transformations.
To calculate $k$, you want to use any pair of $x$ and $y$ from a given dataset. In our example, using the pair $(5,30)$, we find $k$ by rearranging the formula $y=kx$ into $k=\frac{y}{x}$. Thus, $k=\frac{30}{5}=6$. This value of $k$ applies across all data points for the relationship between $y$ and $x$.

Linear Equations

Linear equations are an important concept when dealing with proportional relationships. They express a relationship between two variables, typically $y$ and $x$, in the form of a straight line equation. This form is often expressed as $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept.
In cases of direct proportionality, such as $y=kx$, the line passes through the origin, which means $b=0$.
In the exercise, we have the linear equation $y=6x$. This indicates that for each unit increase in $x$, $y$ increases by 6 units. It's crucial to note here that the graph of this equation would be a straight line passing through the origin (0,0) with a slope of 6.
Understanding how to write and interpret these equations can greatly enhance your ability to predict the behavior of a system based on its linear characteristics.

Experimental Data Analysis

Experimental data analysis involves collecting and interpreting data to understand patterns or relationships. In the context of the exercise, we are given a set of data points that show how $y$ and $x$ relate to each other.
Observing the table, you can see that as $x$ increases, $y$ increases in the same way, indicating a direct proportionality. Detecting this trend involves identifying consistencies within the data.
When you analyze the relationship, calculating the constant of proportionality $k$ from different data points should yield the same result, which confirms the linear relationship. This consistency reassures us that the model $y=6x$ is valid for the given data.
By also using the derived equation, you can predict future outcomes, making experimental data analysis a valuable tool for hypothesis testing and scientific exploration.