Question

Solve $2x - y = 5$ and $3x + 2y = 11$ by substitution.

Answer 1

Hint : To solve a pair of simultaneous linear equations containing two variables, we prefer an algebraic method. An algebraic method is a collection of several methods which are as follows. They are the substitution method, elimination method, and cross-multiplication method. Here, we are asked to solve the given equations by using the substitution method.

Complete step-by-step solution:
The steps that are followed to solve the equations using the substitution method are as follows.
a) We need to solve one of the equations for one variable (either $x$ or $y$)
b) Next, we need to substitute the obtained value from the above step in the other equation.
c) Now, we shall solve the equation to calculate the value of the second variable.
The given equations are $2x - y = 5$ and $3x + 2y = 11$
Here, we are asked to solve the given equations by using the substitution method.
Let $2x - y = 5$…………..$\left( 1 \right)$ and $3x + 2y = 11$………….$\left( 2 \right)$
Now, we shall multiply the equation $\left( 1 \right)$ by $2$
Hence, we get
$2\left( {2x - y = 5} \right) \Rightarrow 4x - 2y = 10$ ………….$\left( 3 \right)$
Now, we need to add the equations $\left( 2 \right)$ and $\left( 3 \right)$
$ \Rightarrow 4x - 2y + 3x + 2y = 10 + 11$
$ \Rightarrow 7x = 21$
$ \Rightarrow x = \dfrac{{21}}{7}$
$ \Rightarrow x = 3$
Now, we shall substitute the value of $x$ in the equation $\left( 1 \right)$
$2x - y = 5 \Rightarrow 2\left( 3 \right) - y = 5$
$ \Rightarrow 6 - y = 5$
$ \Rightarrow y = 6 - 5$
$ \Rightarrow y = 1$
Hence, $\left( {x,y} \right) = \left( {3,1} \right)$ is the required solution.

Note: Here, we are asked to solve the given equations by using the substitution method.
The steps that are followed to solve the equations using the substitution method are as follows.
a) We need to solve one of the equations for one variable (either $x$or$y$)
b) Next, we need to substitute the obtained value from the above step in the other equation.
c) Now, we shall solve the equation to calculate the value of the second variable.