Question

How do you solve the system of equations $- x - 5y = 1$ and $- 3x + 7y = 25$ by combination?

Answer 1

The process of solving any pair of equations by combination method which is also known as the elimination method is done by eliminating any one of the variables by adding or subtracting both the equations. Multiply the first linear equation with the constant before the $x$ variable in the second linear equation and subtract them both to cancel one variable and get a solution for the other.

Complete step by step answer: The given equations are $- x - 5y = 1$ and $- 3x + 7y = 25$
Now consider the first equation, which is $- x - 5y = 1$
Multiply it with the constant in front of the $x$ variable in the second equation $- 3x + 7y = 25$ which is $3$
$\Rightarrow 3( - x - 5y = 1)$
Open the brackets and multiply the constant.
$\Rightarrow - 3x - 15y = 3$
Now subtract both the equations.
$- 3x - 15y = 3$
$( - ) - 3x + 7y = 25$
Now evaluate further,
$\Rightarrow - 3x - ( - 3x) - 15y - 7y = 3 - 25$
Now on further simplification, we get,
$\Rightarrow - 22y = - 22$
Now we get the value of another variable,
$\Rightarrow y = 1$
Now substitute this value of the variable in any of the equation to get the value of another variable $x$
$\Rightarrow - x - 5(1) = 1$
Evaluate further to get,
$\Rightarrow - x = 1 + 5$
$\Rightarrow x = - 6$
$\therefore$ the solution of both the equations upon solving by combination method we get the values of $x,y\;$ as $- 6,1$ respectively.

Additional information: 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.

Note:
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.