Question

How do you solve $y + 6 = 2x$ and $4x - 10y = 4$ ?

Answer 1

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 answer:
First of all, 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.
We need to find the solution of the given two linear equations,
Let us take $y + 6 = 2x$ - - - - - - - - - $(1.)$
and $4x - 10y = 4$ - - - - - - - - - - - - $(2.)$ ,
We need to solve both the equations simultaneously and find a pair of values of the variables.
From $(1.)$ we arrange the terms and write it as $2x - y = 6$ .
Now, multiplying $(1.)$ by $4$ , we get,
$8x - 4y = 24$ - - - - - - - - - - - - $(1.)$
And multiplying $(2.)$ by $2$, we get
$8x - 20y = 8$ - - - - - - - - - - - - $(2.)$ ,
We convert these equations to make the coefficient of $x$ same, same thing can be done with the variable $y$ also.
The new system of equations are
$8x - 4y = 24$ - - - - - - - - - - - - $(1.)$
$8x - 20y = 8$ - - - - - - - - - - - - $(2.)$
Now, we subtract $(2.)$ from $(1.)$ , this becomes
$(8x - 4y) - (8x - 20y) = 24 - 8$
After simplifying the brackets we have
$\Rightarrow 8x - 4y - 8x + 20y = 24 - 8$ ,
Here, the positive and negative of the same term becomes zero.
$\Rightarrow - 4y + 20y = 16$
$\Rightarrow 16y = 16$ ,
Dividing $\;16$ by $\;16$ becomes $1$
$\Rightarrow y = 1$ .
Using this method, we have found the value of one variable. Now, to find the value of another variable, we substitute $y = 1$ in $(1.)$ , then we get from $(1.)$ ,
$8x - 4(1) = 24$
$\Rightarrow 8x - 4 = 24$,
Because $4$ multiplied by $1$ becomes $4$
$\Rightarrow 8x = 24 + 4$
$\Rightarrow 8x = 28$ ,
Taking $8$ on R.H.S. we get
$\Rightarrow x = \dfrac{{28}}{8}$
After simplifying, we get
$\Rightarrow x = \dfrac{7}{2}$ .

$x = \dfrac{7}{2}$ and $y = 1$ , Which is the required solution of the given two linear equations.

Note: A linear equation in two variables has infinitely many solutions. The graph of a linear equation in two variables is a straight line. Every solution of the linear equation in two variables represents a point on the graph of the equation.