Question

How do you solve the system of linear equations \[5x-4y=6\] and $2x+3y=2$ ?

Answer 1

Hint: Now have two equations \[5x-4y=6\] and $2x+3y=2$ . To solve the equation we will first multiply the equations with scalars such that one variable f both equations has the same coefficient. Hence let us say we multiply the first equation by 3 and the second equation by 4. Now we will add the obtained equations and find the value of x. Now substituting the value of x in any equation we will get the value of y.

Complete step by step solution:
Now we are given with two equations \[5x-4y=6\] and $2x+3y=2$ . We know that the equations are linear equations in two variables. Hence we want to find the solution of the system of linear equations. Let us first understand what it means to find the solution of linear equations. Solution of an equation means the values of x and y which will satisfy the equation. Now for one equation in two variables there are infinite solutions. Now if we have two equations we want to find the values of x and y which will satisfy both the equations at once.
Let us understand this concept geometrically.
Now we know that the Linear equation in two variables represents a straight line. The solution of the linear equation means the intersection point of two lines on the equation.
Now consider the given lines \[5x-4y=6\] and $2x+3y=2$ multiplying the first equation by 3 we get,
$15x-12y=18.........\left( 1 \right)$
Now multiplying the equation $2x+3y=2$ by 4 we get,
$8x+12y=8.............\left( 2 \right)$
Now adding equation (1) and (2) on both sides we get,
$23x=26$
Now dividing the whole equation by x we get,
$x=\dfrac{26}{23}$
Now substituting the value of x in equation $2x+3y=2$ we get,
$\begin{align}
  & 2\left( \dfrac{26}{23} \right)+3y=2 \\
 & \Rightarrow 3y=2-\dfrac{52}{23} \\
 & \Rightarrow 3y=\dfrac{46-52}{23} \\
 & \Rightarrow 3y=\dfrac{-6}{23} \\
 & \Rightarrow y=\dfrac{-2}{23} \\
\end{align}$

Hence the solution of the given equation is $x=\dfrac{26}{23}$ and $y=\dfrac{-2}{23}$

Note: Now remember whenever we have a system of linear equations we cannot say that the solutions always exist. Now if we try to understand the concept geometrically we know that the solution of lines is the intersection of lines. Hence we can say that the parallel lines will not have any solution. Hence if the equation represents parallel lines then the equation has no solution.