Question

How‌ ‌do‌ ‌you‌ ‌solve‌ ‌the‌ ‌system‌ ‌$2x+5y=38$‌ ‌and‌ ‌$x-3y=-3$‌ ‌using‌ ‌substitution?‌

Answer 1

Hint: We first rewrite the second equation as $x=-3+3y$ . Then, we put this value of $x$ in the first equation and solve for $y$ . Then, having solved for $y$ , we put this value of $y$ in any one of the equations to get the value of $x$ .

Complete step-by-step solution:
The given equations that we have are
$2x+5y=38....\left( 1 \right)$
$x-3y=-3....\left( 2 \right)$
By substitution, we mean to rearrange one of the equations as a function of one variable like $y=f\left( x \right)$ or $x=f\left( y \right)$ . We then insert this rearranged equation in the other equation to get an equation having only a single variable. This equation can be solved to find the value of the variable. We then put this value of the variable in any one of the equations to get the value of the other variable.
Let us start by first rearranging the equation $\left( 2 \right)$ . We transfer the $3y$ term from the left hand side to the right hand side which gives,
$\Rightarrow x=-3+3y....\left( 3 \right)$
The above equation has been rearranged in the form $x=f\left( y \right)$ . We now insert this function in equation $\left( 1 \right)$ and get,
$2x+5y=38$
$\Rightarrow 2\left( -3+3y \right)+5y=38\,$
Using distributive property, we multiply the $2$ with the terms inside the bracket and get,
$\Rightarrow -6+6y+5y=38\,$
Adding the $y$ terms, we get,
$\Rightarrow -6+11y=38\,$
We add $6$ to both sides of the above equation and get,
$\Rightarrow 11y=38\,+6$
Adding the arithmetic terms, we get,
$\Rightarrow 11y=44$
Dividing both sides of the above equation by $11$ , we get,
$\Rightarrow \dfrac{11y}{11}=\dfrac{44}{11}$
Simplifying the above equation, we get,
$\Rightarrow y=4$
We now put this value of $y$ in the equation $\left( 3 \right)$ to get,
$\Rightarrow x=-3+3\left( 4 \right)$
$\Rightarrow x=9$
Therefore, we can conclude that the solution of the given system of equations is $x=9,y=4$ .

Note: We should be careful while rearranging the equation as a function of a single variable, by taking care of the signs. Also, when we substitute the function, the expression gets very large and thus most susceptible to mistakes. In the end, we should cross check our answer by putting the values in both the given equations.