135,698 views
32 votes
32 votes
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?

User MikeCW
by
2.0k points

1 Answer

20 votes
20 votes

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

x

User Kotatsuyaki
by
2.8k points