Question

How do you solve the system of equations $x + y = 10$ and $y = x - 6$ ?

Answer 1

Hint: We have given two linear equations in two variables. Now, to solve both the equations, we need to solve both the equations simultaneously and hence we will find the value of the two variables in the linear equations.

Complete step-by-step solution:
The equations of the form $ax + by + c = 0$ or $ax + by = c$ , where $a$ , $b$ and $c$ are real numbers, $a \ne 0$ , $b \ne 0$ and $x$ , $y$ are variables, is called a linear equation in two variables.
Now, to find the solution of the given two linear equations,
We take $x + y = 10$ - - - - - - - - - $(1.)$
and $y = x - 6$ - - - - - - - - - - - - $(2.)$
which means that we need to solve both the equations simultaneously and find a pair of values of the variables, which will be the required solution of the given linear equations.
From $(2.)$ we arrange the terms and write it as $x - y = 6$ .
Now, by adding $(1.)$ and $(2.)$ , we get,
$x + y + x - y = 10 + 6$
$\Rightarrow 2x = 16$ ,
Dividing by $2$ on both sides, we get
$\Rightarrow \dfrac{{2x}}{2} = \dfrac{{16}}{2}$
$\Rightarrow x = 8$ .
Using this method, we have found the value of one variable. Now, to find the value of another variable, we substitute $x = 8$ in $(1.)$ , then we get from $(1.)$ ,
$8 + y = 10$
$\Rightarrow y = 10 - 8$
Hence, simplifying further
$\Rightarrow y = 2$ .

$x = 8$ and $y = 2$ , which is the required solution of the given two linear equations.

Note: The values of the variables of a linear equation is the solution of the linear equation. A straight line on a graph is the representation of a linear equation. Every point on the line represents the solution of the linear equation.