6 Find Y and C from the following Y C I0 G0 C 25 6y I0

6. Find Y* and C* from the following:

            Y = C + I0 + G0

            C = 25 + 6y½

            I0 = 16

             G0 = 18

Solution

Y = C + I0+ G0

Substituting C, I0, G0 values

Y = 25 + 6Y^1/2 + 16 + 18

Y = 6Y^1/2 + 59

Y - 59 = 6Y^1/2

Squaring both sides

(Y - 59)² = (6Y^1/2)²

Y² - 118Y + 3481 = 36Y

Y² - 118Y - 36Y + 3481 = 0

Y² - 154Y + 3481 = 0

Y = (-(-154) + or - sqrt((-154)² - 4 * 1 * 3481))/2

Y = (154 + or - sqrt(9792))/2

Y = 77 + or - 49.48

Y = 77 + 49.48 or Y = 77 - 49.48

Y = 126.48 or Y = 27.52

Substituting Y = 126.48 to find C

C = 25 + 6Y^1/2

C = 25 + 6*sqrt(126.48)

C = 25 + 6*(11.246) or C = 25 + 6*(-11.246)

C = 92.48 or C= -42.48

Substituting Y = 126.48 & C = 92.48 values in Original Equation

Y = C + I0 + G0

126.48 = 92.48 + 16 + 18

126.48 = 126.48

This satisfies the equation.

So values of Y & C are Y = 126.48 and C = 92.48

Substituting Y = 126.48 & C = -42.48 values in Original Equation

Y = C + I0 + G0

126.48 = -42.48 + 16 + 18

126.48 = -8.48

This does not satisfy the equation.

Substituting Y = 27.52 to find C

C = 25 + 6Y^1/2

C = 25 + 6*sqrt(27.52)

C = 25 + 6*(5.246) or C = 25 + 6*(-5.246)

C = 56.48 or C= -6.48

Substituting Y = 27.52 & C = 56.48 values in Original Equation

Y = C + I0 + G0

27.52 = 56.48 + 16 + 18

27.52 = 90.52

This does not satisfy the equation.

Substituting Y = 27.52 & C = -6.48 values in Original Equation

Y = C + I0 + G0

27.52 = -6.48 + 16 + 18

27.52 = 27.52

This satisfies the equation.

So the values are Y = 27.52 and C = -6.48

The values of Y* and C* are

Y = 126.48 and C = 92.48 or

Y = 27.52 and -6.48