Question

How do you solve the system $4x-7y=9$ and $y=x-3$ using substitution?

Answer 1

Hint: The process of solving any pair of equations by substitution method is done by eliminating any one of the variables by substituting either of the variables in the other equation. First, we take any equation and write the equation in terms of one of the variables and then substitute the same equation in the second equation. One of the variable’s values will be then evaluated. Substitute the value of it in the first equation or second to get the other variable too.

Complete step by step solution:
The given equations are $4x-7y=9$ and $y=x-3$
Now consider the equation, $y=x-3$
We have considered this equation because in the substitution method we first need to take an equation and then write it in terms of one variable which would be later used to substitute in another equation.
The above equation is already written in terms of $y$ hence we use this equation to substitute in the second equation which is $4x-7y=9$
The given equation in terms of $y$ is $y=x-3$ , we substitute this in the second equation which is $4x-7y=9$
$\Rightarrow 4x-7\left( x-3 \right)=9$
Now let us evaluate to find the value of $x$
$\Rightarrow 4x-7x+21=9$
$\Rightarrow -3x=9-21$
Further, evaluate.
$\Rightarrow -3x=-12$
Divide the entire equation with -3
$\Rightarrow \dfrac{-3x}{-3}=\dfrac{-12}{-3}$
$\Rightarrow x=4$
Now substitute this in any of the equation to get the value of $y$
$\Rightarrow y=x-3$
$\Rightarrow y=4-3$
$\Rightarrow y=1$

$\therefore$ The solution of both the equations upon solving by substitution method we get the values of $x,y\;$ as $4,1\;$ respectively.

Note: Any pair of linear equations can be solved by using three methods which are,
The elimination method, the cross-multiplication method, and the substitution method. In the elimination or combination method, we add or subtract both the equations to eliminate one variable and find the solution of others and then substitute and get the solution of the other one also. In the substitution method, we represent one variable in the form of another and substitute it in the other equation to get the solution.
This question can also be solved by the cross-multiplication method which is, we calculate using the formula, $x=\dfrac{{{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}};y=\dfrac{{{c}_{1}}{{a}_{2}}-{{c}_{2}}{{a}_{1}}}{{{a}_{1}}{{b}_{2}}-{{a}_{2}}{{b}_{1}}}$ .Write the equations in general form and then substitute the values to get the solution for both the equations.