Algebraic Expressions

Given$(x,y)=(p,q)$
$(3x - 1, 9) = (11, p + 2)$
By equating 
$3x - 1 = 11$
$3x = 12$
$x = 4$
$9 = p + 2$
$p = 7$
So, the value of $x = 4, p = 7.$

Given $(x,y)=(p,q)$
$(4y + 5, 3p - 1) = (25, p + 1)$
On equating we get
$4y + 5 = 25$
$4y = 25 - 5$
$4y = 20$
$y = 5$
$3p - 1 = p + 1$
$3p - p = 1 + 1$
$2p = 2$
$p = 1$
So, the value of$ p = 1, y = 5$

as the given ordered pairs are equal,

${ \left( x-y \right)  }^{ 2 }+{ x }^{ 2 }=25$
Now, ${ 3 }^{ 2 }+{ 4 }^{ 2 }={ 5 }^{ 2 }$
$9+16=25$
$\therefore $There are 2 possibilities:
$I.{ \left( x-y \right)  }^{ 2 }=9$ and ${ x }^{ 2 }=16$
$\therefore x=\pm 4$ and $x-y=\pm 3$
$\left( i \right) .x-y=3\Rightarrow \left( 4,1 \right) $ and $\left( -4,-7 \right) $
$\left( ii \right) .x-y=-3\Rightarrow \left( 4,7 \right) $ and $\left( -4,-1 \right) $
$II.{ \left( x-y \right)  }^{ 2 }=16$ and ${ x }^{ 2 }=9$
$\therefore x=\pm 3$ and $x-y=\pm 4$
$\left( i \right) .x-y=4\Rightarrow \left( 3,-1 \right) $ and $\left( -3,-7 \right) $
$\left( ii \right) .x-y=-4\Rightarrow \left( 3,7 \right) $ and $\left( -3,-1 \right) $
$\therefore $ Different values of y are $1,-1,7,-7$
$\therefore 4$ different values of y occur.

The regression coefficient of Y on X is given as 0.5. Rewriting the equation 2Y = aX - 16.80 in slope form gives Y = (a/2)X - 8.40. The slope (regression coefficient) equals a/2 = 0.5, therefore a = 1.0.

The correlation coefficient (r) is the geometric mean of the two regression coefficients (b_yx and b_xy). Here, r = sqrt(0.49 * 1) = sqrt(0.49) = 0.7.

The correlation coefficient (r) is the geometric mean of the regression coefficients. Since both coefficients are negative, r must also be negative: r = -sqrt((-1/3) * (-3/4)) = -sqrt(1/4) = -0.5.

The regression coefficient bxy is defined as the slope of the regression line of X on Y. In the equation 2.9X = aY + 15, we can rewrite it as X = (a/2.9)Y + (15/2.9). Since bxy = a/2.9 and bxy = 2.0, we have a/2.9 = 2.0, which means a = 5.8.

The correlation coefficient r is the geometric mean of the two regression coefficients, calculated as r = sqrt(bxy * byx). Here, r = sqrt(0.25 * 0.64) = sqrt(0.16) = 0.40.

Calculating the Pearson correlation coefficient for the given data points (21,10), (24,14), (28,18), (30,20), (32,23) yields a value very close to 1, indicating a strong positive linear relationship.

The correlation coefficient r is the geometric mean of the regression coefficients: r = sqrt(0.8 * 0.2) = sqrt(0.16) = 0.40. Since the regression coefficients are positive, the correlation coefficient is positive.

The correlation coefficient r is the geometric mean of the two regression coefficients (b_xy and b_yx). Here, r = sqrt((-2/3) * (-1/6)) = sqrt(2/18) = sqrt(1/9) = -1/3 (the sign matches the regression coefficients).

The correlation coefficient r is the square root of the product of the regression coefficients. Here, r = sqrt(0.64 * 1) = sqrt(0.64) = 0.8.

Put x = 3 and y = 3 of LHS, we get
5$\dfrac{1}{3}$ $\times$ 3$\dfrac{3}{4}$ = $\dfrac{16}{3}$ $\times$ $\dfrac{15}{4}$ = 4 $\times$ 5 = 20