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

Hint: A set of simultaneous equations, which is also known as a system of equations or an equation system, is a finite set of equations representing some mathematical expressions, situations.
A system of equations or an equation system having one or two variables consist of a common solution. A system of equations can be a system of linear equations or a system of linear equations.
System of linear equations: A system of linear equations is a collection of one or more linear equations having the same set of variables.
A solution to a linear equation is an assignment of values to the variables such that all the equations are simultaneously satisfied. to solve the system of linear equations use elimination method.
Elimination method: Multiply both the given equations by a non-zero constant to make the coefficients of any variable numerically equal, then add or subtract so that equal coefficients of one variable get eliminated. Now solve the equations in one variable and get the value. Substitute the value of one variable into another equation and solve the equation.
System of nonlinear equations: A system of nonlinear equations is a set of equations in which unknown variables are commonly approximated by linear equations.

Complete step-by-step answer:
Step: 1 Let the first number is $x$ and the second number is $y$.
According to question,
The sum of two numbers is equal to 32 therefore,
$x + y = 32 \ldots \ldots \ldots \left( 1 \right)$
The difference between the numbers is 8, so
$x - y = 8 \ldots \ldots \ldots \left( 2 \right)$
Step: 2 Multiply both equations by $\left( 1 \right)$.
$\left( {x + y} \right) \times 1 = \left( {1 \times 32} \right) \ldots \ldots \ldots \left( 1 \right)$
Multiply the equation 2 by 1.
$\left( {x - y} \right) \times 1 = \left( {1 \times 8} \right) \ldots \ldots \ldots \left( 2 \right)$
Step: 3 Add the both equations two eliminate one variable and solve,
$
  x + y = 32 \ldots \ldots \ldots \ldots \left( 1 \right) \\
  \underline {x - y} = 8 \ldots \ldots \ldots \ldots \left( 2 \right) \\
   \Rightarrow 2x = 40 \\
 $
Divide the equation both side with 2,
$
  \dfrac{{2x}}{2} = \dfrac{{40}}{2} \ldots \ldots \ldots \ldots \ldots \left( 3 \right) \\
   \Rightarrow x = 20 \\
 $
Step: 4 Substitute the value of $\left( {x = 20} \right)$ in the first equation.
$
  x + y = 32 \ldots \ldots \ldots \left( 1 \right) \\
   \Rightarrow 20 + y = 32 \\
   \Rightarrow y = 32 - 20 \\
   \Rightarrow y = 12 \\
 $
Therefore the first number is 20 and the second number is 12.
Therefore the required system of equations to represent the situation is,
$x + y = 32 \ldots \ldots \ldots \left( 1 \right)$ and $x - y = 8 \ldots \ldots \ldots \left( 2 \right)$

Therefore, the first number is 20 and the second number is 12.

Note:
Students are advised to, not make any mistake to write the system of equations correctly.
They must know the elimination method to solve the system of equations involving the two variables.
Step: 1 Multiply each equation by a suitable number so that the two equations have the same leading coefficient.
Step: 2 Add both equations two eliminate one variable. They must apply the elimination method step by step, carefully to avoid any mistake which is done by most of the students.