Question

If the two lines of regressions are \[4x + 3y + 7 = 0\]and \[3x + 4y + 8 = 0\] then the means of \[x\] and \[y\]are
A). \[ - \dfrac{4}{7}, - \dfrac{{11}}{7}\]
B). \[ - \dfrac{4}{7},\dfrac{{11}}{7}\]
C). \[\dfrac{4}{7}, - \dfrac{{11}}{7}\]
D). \[4,7\]

Answer 1

Hint: Here we are given the regressions of two lines that have been given and asked to find the means of the terms \[x\] and\[y\]. The measure of the average relationship between two or more variables is known as regression. The means of the unknown variables \[x\]and \[y\] can be found by solving the given regression equations of the lines since they lie in that.

Complete step-by-step solution:
It is given by the regression of two lines, \[4x + 3y + 7 = 0\] and \[3x + 4y + 8 = 0\]. We aim to find the mean values of the variables \[x\] and \[y\].
We know that the means of the unknown variable lie in the equation of regression of the line.
So, by solving the given equation of regression we can find the mean values of the unknown variables \[x\] and \[y\].
Now let us start to solve the given equations. Consider the equations
\[4x + 3y + 7 = 0.......(1)\]
\[3x + 4y + 8 = 0.......(2)\]
On solving \[(1)\] and \[(2)\] we get
\[(1) \times 3 \Rightarrow 12x + 9y + 21 = 0.......(3)\]
\[(2) \times 4 \Rightarrow 12x + 16y + 32 = 0........(4)\]
Now subtracting the equation\[(4)\] from\[(3)\] we get
\[ \Rightarrow 12x + 9y + 21 - 12x - 16y - 32 = 0\]
Now let us multiply the minus sign into the equation\[(4)\].
\[ \Rightarrow 12x + 9y + 21 - 12x - 16y - 32 = 0\]
Now let us group the like terms and simplify them. On simplifying this we get
\[ \Rightarrow - 7y - 11 = 0\]
On further simplification we get
 \[
   \Rightarrow - 7y = 11 \\
   \Rightarrow y = - \dfrac{{11}}{7} \]
Thus, we got the value of one variable, let’s find the other by substituting this into the equation\[(1)\].
\[(1) \Rightarrow 4x + 3\left( { - \dfrac{{11}}{7}} \right) + 7 = 0\]
On simplifying this we get
\[ \Rightarrow 4x - \dfrac{{33}}{7} + 7 = 0\]
\[ \Rightarrow 4x = \dfrac{{33}}{7} - 7\]
\[ \Rightarrow 4x = - \dfrac{{16}}{7}\]
\[ \Rightarrow x = - \dfrac{4}{7}\]
Thus, we got the values of both variables. Hence, the mean values of the variables are \[x = - \dfrac{4}{7}\]and\[y = - \dfrac{{11}}{7}\].
Now let us see the options, option (a) \[ - \dfrac{4}{7}, - \dfrac{{11}}{7}\]is the correct option as we got the same values in our calculations.
Option (b) \[ - \dfrac{4}{7},\dfrac{{11}}{7}\]is an incorrect option as we got \[x = - \dfrac{4}{7}\]and\[y = - \dfrac{{11}}{7}\].
Option (c) \[\dfrac{4}{7}, - \dfrac{{11}}{7}\]is an incorrect option as we got \[x = - \dfrac{4}{7}\]and\[y = - \dfrac{{11}}{7}\].
Option (d) \[4,7\]is an incorrect option as we got \[x = - \dfrac{4}{7}\]and\[y = - \dfrac{{11}}{7}\].
Hence, option (a) \[ - \dfrac{4}{7}, - \dfrac{{11}}{7}\] is the correct answer.

Note: Here while doing simplifications we have grouped like terms together. Like terms are nothing but the terms having the same unknown variable raised to the same powers. Addition and subtraction of algebraic expressions are done by grouping like terms and simplifying them.