Question

the product of the ages of Lisa and her twin bothers is 36, and the sum of their ages is 13. How old is Lisa?

Answer 1

9 years

Step-by-step explanation

Step 1

set the equations.

let

x represents the lisa's age

y represents the twin bother's age

so

a)the product of the ages of Lisa and her twin bothers is 36

hence

`xy=36\Rightarrow equation(1)`

b)and the sum of their ages is 13

`x+y=13\Rightarrow equation(2)`

Step 2

solve the equations

a) isolate the y value in equation (2) then replace in eqaution (1)

`\begin{gathered} x+y=13\Rightarrow equation(2) \\ \text{subtract x in both sides} \\ x+y-x=13-x \\ y=13-x \end{gathered}`

now , replace in eq(1)

`\begin{gathered} xy=36\Rightarrow equation(1) \\ x(13-x)=36 \\ 13x-x^2=36 \\ \text{subtract 36 in both sides an reorder} \\ 13x-x^2-36=36-36 \\ -x^2+13x-36=0 \end{gathered}`

we need to solve this quadratic equation, let's use the quadratic formula

`\begin{gathered} \text{for ax}^2+bx+c=0 \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \end{gathered}`

hence

`\begin{gathered} -x^2+13x-36=0\Rightarrow ax^2+bx+c \\ so \\ a=-1 \\ b=13 \\ c=-36 \end{gathered}`

now, to find the solutino for x, let's replace in the formula

`\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x=\frac{-(13)\pm\sqrt[]{13^2-4(-1)(-36)}}{2(-1)} \\ x=\frac{-(13)\pm\sqrt[]{169-144}}{-2} \\ x=\frac{-(13)\pm\sqrt[]{25}}{-2} \\ x=(-(13)\pm5)/(-2) \end{gathered}`

we have the symbol

`\pm`

it means, there are two solutions, let's check

`\begin{gathered} x=(-(13)\pm5)/(-2) \\ x_1=(-(13)+5)/(-2)=(-8)/(-2)=4 \\ x_2=(-(13)-5)/(-2)=(-18)/(-2)=9 \end{gathered}`

so,

`-x^2+13x-36=0=(x-4)(x-9)`

Step 3

let's solve the quadratic equation by factoring

`\begin{gathered} -x^2+13x-36=0 \\ \text{change the signs} \\ x^2-13x+36=0 \\ \text{rewrite -13x as -4x-9x} \\ x^2-4x-9x+36=0 \\ \text{factorize} \\ x(x-4)-9(x-4)=0 \\ (x-4)(x-9)=0 \end{gathered}`

so, the posibles values for Lisa´s age are

9 or 4

as we don't know who is older, the ages are 4 and 9

let's prove

a)the product of the ages of Lisa and her twin bothers is 36

`9\cdot4=36`

b)

and the sum of their ages is 13.

`9+4=13`

I hope this helps you

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