Question

If \(x\) and \(y\) are related as \(4 x-3 y=10\) and the mean deviation of \(\mathrm{x}\) is 10 then the mean deviation of \(\mathrm{y}\) is (a) 13 (b) \(12.3\) (c) \(13.3\) (d) \(13.5\)

Short answer

The mean deviation of y is 13.3.

Step-by-step solution

Find the relationship between the mean deviations of x and y

Since \(4x - 3y = 10\), we can say that \(y = \frac{4}{3}x - \frac{10}{3}\). Let the mean deviation of x be \(M_x\) and the mean deviation of y be \(M_y\). Let \(x_1\) and \(x_2\) be any two values of x such that their mean deviation is 10. Then, \(M_x = |x_1 - x_2| = 10\) Now, we need to find the corresponding values of y for these x values, i.e., \(y_1\) and \(y_2\). We can do that using the relationship we found. \(y_1 = \frac{4}{3}x_1 - \frac{10}{3}\) \(y_2 = \frac{4}{3}x_2 - \frac{10}{3}\)

Calculate the mean deviation of y using the relationship

Now, let's find the mean deviation of y using the values of \(y_1\) and \(y_2\): \(M_y = |y_1 - y_2|\) Now substitute the values of \(y_1\) and \(y_2\) we found in step 1: \(M_y = \left|\left(\frac{4}{3}x_1 - \frac{10}{3}\right) - \left(\frac{4}{3}x_2 - \frac{10}{3}\right)\right|\) Simplify the expression: \(M_y = \left|\frac{4}{3}(x_1 - x_2)\right|\) Since we know \(|x_1 - x_2| = M_x = 10\), we can substitute it in the expression: \(M_y = \frac{4}{3}(10)\)

Find the mean deviation of y

Multiply the fraction and the constant to find the mean deviation of y: \(M_y = 4\left(\frac{10}{3}\right) = \frac{40}{3} = 13.3\) So, the mean deviation of y is \(13.3\). Therefore, the correct answer is (c) 13.3.

Key concepts

Linear Equations

Linear equations form the backbone of algebra. They are equations that represent a straight line when plotted on a graph. In essence, they take the form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. This form is known as the standard form of a linear equation.

For our exercise, we have the linear equation \(4x - 3y = 10\). It describes a relationship between two variables \(x\) and \(y\). Solving such equations involves finding values for \(x\) and \(y\) that satisfy the equation simultaneously. You can visually interpret the equation as a line on a graph, where every point on the line is a solution that meets the condition of the equation.

• Linear equations can be transformed into slope-intercept form \(y = mx + b\), which highlights the slope \(m\) and the y-intercept \(b\) of the line.
• They can also be used to describe relationships between variables, as in real-life scenarios like economics, physics, and various fields of science.

Here, the equation was rearranged to express \(y\) as dependent on \(x\): \(y = \frac{4}{3}x - \frac{10}{3}\). This rearranged form allows us to understand how changes in \(x\) will affect \(y\).

Mean Deviation Calculation

Mean deviation is a measure of dispersion that shows how much the values in a data set differ from the average of the set. It is a straightforward way to grasp the idea of spread or variability. The mean deviation is calculated by averaging the absolute differences between each value in the set and the mean of the set.

• The formula for mean deviation is: \(M = \frac{1}{n} \sum |x_i - \bar{x}|\) where \(x_i\) are the data points and \(\bar{x}\) is the mean.
• In practice, mean deviation helps us understand how "spread out" the data points in a certain set are.

In the problem, \(M_x = 10\) represents the mean deviation of \(x\). This means that, on average, the values of \(x\) deviate from the mean by 10 units.

To find the mean deviation of \(y\), we utilized the linear relationship between \(x\) and \(y\). Substituting the expression for \(y\) into the deviation formula gives \(M_y = \left|\frac{4}{3}(x_1 - x_2)\right|\). Given that the mean deviation for \(x\) \(|x_1 - x_2| = 10\), we subsequently calculated \(M_y = \frac{40}{3} = 13.3\).

Variable Transformation

Variable transformation refers to the process of changing variables within equations to find new relationships or simplify calculations. This technique is frequently used when dealing with problems that involve linear equations or converting one form of an equation to another.

In the exercise, we started with the equation \(4x - 3y = 10\) and transformed it by expressing \(y\) in terms of \(x\): \(y = \frac{4}{3}x - \frac{10}{3}\). Such a transformation made it easier to relate and compute the mean deviation between \(x\) and \(y\).

• Variable transformation is useful when solving systems of equations, allowing us to simplify the problem into a single-variable equation.
• It is also beneficial in contexts where one seeks to understand how a change in one variable affects another to draw conclusions or make predictions.

By transforming \(x\) into \(y\), the exercise further illustrates how transformations can be leveraged to analytically solve for intricate requirements, like differing deviations among linked variables.