Question

Solve \[8x + 5y = 9\] , \[3x + 2y = 4\] by any method.

Answer 1

Hint:In this given question, we have to solve the system of linear equations in two variables. We will use a substitution method to find a solution. We will first transform the first equation (i.e. \[8x + 5y = 9\] ) to find the value of \[y\] . And then substitute this value of \[y\] into the second equation (i.e. \[3x + 2y = 4\] ). On solving we get the value and hence we can find the value from any other equation.

Complete step by step answer:
This is based on a system of linear equations in two variables. An equation of the form in which the highest power of each variable is one is called a linear equation in two variables.Solution of the system of linear equations is the value that satisfies the equation.Consider the given question,
The given equation are:
\[8x + 5y = 9\] ……………………(1)
\[\Rightarrow 3x + 2y = 4\] …………………(2)
From equation (1), we have, \[8x + 5y = 9\]
Adding \[ - 8x\] both sides and then dividing by \[5\] , we get
\[y = \dfrac{{9 - 8x}}{5}\]
Putting the value of \[y\] in equation (2), we get
\[3x + 2y = 4 \\
\Rightarrow 3x + 2\left( {\dfrac{{9 - 8x}}{5}} \right) = 4 \\ \]

On talking LCM, we have
\[ \Rightarrow \dfrac{{15x + 2(9 - 8x)}}{5} = 4\]
Multiplying both side by \[5\], we get
\[15x + 2(9 - 8x) = 20 \\
\Rightarrow 15x + 18 - 16x = 20 \\ \]
Adding \[ - 18\] both side we get
\[- x = 2\]
Multiplying both side by \[ - 1\] we get,
Hence, \[x = - 2\]
Putting the value of \[x = - 2\] in \[y = \dfrac{{9 - 8x}}{5}\] , we have
\[ y = \dfrac{{9 - 8( - 2)}}{5} \\
\Rightarrow y = \dfrac{{25}}{5} \\
\therefore y = 5\]

Hence \[x = - 2\] and \[y = 5\] is the required solution.

Note: The system of linear equations can also be solved using elimination method and method of Cross multiplication. In the substitution method, we substitute the value of one variable from the first equation into the second equation. In the elimination method, we eliminate the variable by adding, subtracting the equation. We need at least two equations to solve problems on linear equations in two variables.