Question

The sum of two numbers is $32$. The difference between the numbers is $8$. How do you write a system of equations to represent this situation and solve?

Answer 1

In this question, they have given two statements and we need to write an equation and solve it accordingly. First we will form an equation as per given in the question and by using a substitution method, we are able to solve the equations easily.

Complete step by step answer:
In this question it is said that the sum of two numbers is $32$.
First let us assume the two different numbers as $a$ and $b$.
So this implies that the sum of $a$ and $b$ is $32$.i.e. $a + b = 32$
It is also said that the difference between the numbers is$8$.
Then the difference between $a$ and $b$ is $8$ .
Therefore $a - b = 8$
Now we got the two different equations which are: $a + b = 32$ and $a - b = 8$.
Now we need to find the two numbers. For that we will use a substitution method. As we can see there are two simple equations, we can substitute the only of one in the other.
Taking the second equation $a - b = 8$,
We can derive $a$ by transferring $b$ to the other side,
$\Rightarrow a = 8 + b$
So as we got $a$ from the second equation, we can substitute this value in the first equation.
Substituting $a = 8 + b$ in $a + b = 32$ we get,
$\Rightarrow a + b = 32$
$\Rightarrow 8 + b + b = 32$
Adding the variables,
$\Rightarrow 8 + 2b = 32$
Now, transferring $8$ to the other side,
$\Rightarrow 2b = 32 - 8$
On subtracting we get,
$\Rightarrow 2b = 24$
Dividing $2$ on both sides we get,
$\Rightarrow b = \dfrac{{24}}{2}$
Therefore the value of $b$ is,
$\Rightarrow b = 12$
As we got the value of $b$, we will now substitute this in $a = 8 + b$ and find the value of $a$.
$a = 8 + b$
$\Rightarrow a = 8 + 12$
$\Rightarrow a = 20$

Therefore, the values of $a$ and $b$ are $20$ and $12$ .

Note: Alternative method:
We can find the values of $a$ and $b$ by another method also,
As we know the two equations, $a + b = 32$ and $a - b = 8$
We can either add or subtract it in order to get the value of any one of the variables.
We have to do it in such a way that one variable should cancel out itself. We can alter or rewrite the equations for that purpose.
Now adding the both equation, we get
${a + b = 32} \\\ {\underline {{\text{ }}a - b = 8{\text{( + )}}} {\text{ }}} \\\ {{\text{ }}2a = 40}$
As we can see there are two positive $a$ so it becomes $2a$ when added. And there is one $+ b$ and one $- b$ , so when they are added, they cancel out and become $0$ . In this way we can find the value of other variables.
So now, finding $a$ ,
$\Rightarrow a = \dfrac{{40}}{2}$
We get, $a = 20$
Now substituting the value of $a$ in any of the equations, we can find $b$ .
Let’s put $a = 20$ in the second equation.
$\Rightarrow 20 - b = 8$
$\Rightarrow b = 20 - 8$
Let us subtract the terms and we get,
$\Rightarrow b = 12$
Therefore the values of $a$ and $b$ are $20$ and $12$ .