Question

How do you solve by substitution $3x-2y=11\And x-\dfrac{1}{2}y=4$?

Answer 1

Hint: We will solve this question using substitution method. We have been given two sets of equations with two variables namely $x$ and $y$. So, we will take the equation $x-\dfrac{1}{2}y=4$ and re-write the equation in terms of $x$. Then, we will substitute the value of $x$ in the other equation given, which is, $3x-2y=11$ and we will obtain the value of $y$. Now, we will substitute the value of $y$ in the first equation which we wrote in terms of $x$, to get the true value of $x$.

Complete step by step solution:
According to the given question, we have been provided with two sets of equations with two variables and we have to find the value of the variables namely $x$ and $y$ using a substitution method only.
We will start by taking an equation and writing it in terms of one of the variables. We have,
$x-\dfrac{1}{2}y=4$
We can rearrange it as,
$\Rightarrow x=4+\dfrac{1}{2}y$---------(1)
Now, substituting the value of $x$ in the other we have, so we get,
$3x-2y=11$---------(2)
$\Rightarrow 3\left( 4+\dfrac{1}{2}y \right)-2y=11$
Opening up the brackets and multiplying the terms, we get,
$\Rightarrow 3(4)+3\left( \dfrac{1}{2}y \right)-2y=11$
$\Rightarrow 12+\left( \dfrac{3}{2}y \right)-2y=11$
Separating the y-components and the constants, we have,
$\Rightarrow 2y-\left( \dfrac{3}{2}y \right)=12-11$
In the LHS we have the y-components, we will take the LCM of the terms and solve the above expression, we have,
$\Rightarrow \dfrac{4y}{2}-\left( \dfrac{3}{2}y \right)=12-11$
$\Rightarrow \dfrac{4y-3y}{2}=1$
We now have,
$\Rightarrow \dfrac{y}{2}=1$
Now, multiplying 2 on both the sides, we get,
$\Rightarrow y=2$
So we have the value of y, we substitute this value in equation (1), we get,
$\Rightarrow x=4+\dfrac{1}{2}(2)$
$\Rightarrow x=4+1$
$\Rightarrow x=5$
Therefore, we have $x=5$ and $y=2$.

Note: When writing the equation in terms of one of the variable, be careful not to substitute the value of that variable in the same equation because then all the terms will be similar and will get cancelled. Also, while substituting the values, calculate the values of each variable step-wise.