Question

How do you solve the system of equations $6x + 5y = 10$ and $7x - 3y = 24$?

Answer 1

Hint: In this question we have to solve the system of equations, and this can be done by using elimination method, and in elimination method first multiply first equation by 3 and second equation with 5 so that the two equations have the same leading coefficient, then subtract second equation from the first then solve the equations for the value of $x$, now substitute the $x$ value in any of the equations to get the value of $y$.

Complete step by step solution:
Elimination method refers to the addition method of solving a set of linear equations. This is quite similar to the method that we would have learned for solving linear equations.
Given system of equations are $6x + 5y = 10$ and $7x - 3y = 24$,
$ \Rightarrow 6x + 5y = 10 - - - - (1)$
And $7x - 3y = 24 - - - - (2)$,
Now multiply 3 to the first equation and multiply 2 to the second equation, we get,
$ \Rightarrow 6x + 5y = 10 - - - - (1) \times 3$
$7x - 3y = 24 - - - - (2) \times 5$,
Now simplifying we get,
$ \Rightarrow 18x + 15y = 30 - - - - (1)$
$35x - 15y = 120 - - - - (2)$,
 Now adding second equation and the first equation we get,
$ \Rightarrow 18x + 15y = 30 - - - - (1)$
$35x - 15y = 120 - - - - (2)$,
Now simplifying we get,
$ \Rightarrow 18x + 35x = 30 + 120$,
Now simplifying we get,
$ \Rightarrow 53x = 150$,
Now divide both sides of the equation with 53 we get,
$ \Rightarrow \dfrac{{53x}}{{53}} = \dfrac{{150}}{{53}}$,
Now simplifying we get,
$ \Rightarrow x = \dfrac{{150}}{{53}}$,
Now substituting the value of $x$ in the first equation, we get,
$ \Rightarrow 6\left( {\dfrac{{150}}{{53}}} \right) + 5y = 10$
Now taking out the common term from all terms we get,
$ \Rightarrow 6\left( {\dfrac{{30}}{{53}}} \right) + y = 2$,
Now multiplying we get,
$ \Rightarrow \dfrac{{180}}{{53}} + y = 2$,
Now subtract $\dfrac{{180}}{{53}}$ from both sides of the equation we get,
$ \Rightarrow \dfrac{{180}}{{53}} + y - \dfrac{{180}}{{53}} = 2 - \dfrac{{180}}{{53}}$,
Now simplifying we get,
$ \Rightarrow y = \dfrac{{106 - 180}}{{53}}$,
Now simplifying we get,
$ \Rightarrow y = \dfrac{{ - 74}}{{53}}$,
So, the values of $x$ and $y$ are $\dfrac{{150}}{{53}}$ and $\dfrac{{ - 74}}{{53}}$.

$\therefore $The value of $x$ and $y$ when the system of equations $6x + 5y = 10$ and $7x - 3y = 24$ are solved are $\dfrac{{150}}{{53}}$ and $\dfrac{{ - 74}}{{53}}$.

Note: The three methods which are most commonly used to solve systems of equations are substitution, elimination and augmented matrices.
Substitution and elimination are simpler methods of solving equations and are used much more frequently than augmented matrices in basic algebra. The substitution method is especially useful when one of the variables is already isolated in one of the equations. The elimination method is useful when the coefficient of one of the variables is the same (or its negative equivalent) in all of the equations. The primary advantage of augmented matrices is that it can be used to solve systems of three or more equations in situations where substitution and elimination are either unfeasible or impossible.