Question

How do you solve the system of equations \[x + 3y = - 4\] and \[5x + 4y = 13\] ?

Answer 1

Hint:In this question, we need to solve the system of given equations \[x + 3y = - 4\] and \[5x + 4y = 13\]. On observing the given equations , both are linear equations with two variables . We can solve them by using a substitution method. First we need to express one of the variables \[x\] or \[y\] in the terms of the other variable from either of any two equations . Then we need to substitute that expression in the other equation to solve for which variable is left.

Complete step by step solution:
Given, \[x + 3y = - 4\] and \[5x + 4y = 13\]
Here we need to solve the system of equation \[x + 3y = - 4\] and \[5x + 4y = 13\]
Let us consider \[x + 3y = - 4\] ••• (1) and \[5x + 4y = 13\] ••• (2)
First we can solve the equation (1) for \[x\] because we can see smaller numbers that are easier to deal with.
Let us consider equation (1)
\[\Rightarrow \ x + 3y = - 4\]
By subtracting \[3y\] on both sides to get \[x\] isolated ,
We get,
\[\Rightarrow \ x = - 4 – 3y\] ••• (3)
Now let us consider equation (2) ,
\[\Rightarrow \ 5x + 4y = 13\]
On substituting \[x\] in equation (2),
We get ,
\[\Rightarrow \ 5( - 4 – 3y)\ + 4y = 13\]
On multiplying the terms inside,
We get,
\[\Rightarrow \ - 20 – 15y + 4y = 13\]
On adding like terms,
We get,
\[\Rightarrow \ - 20 – 11y = 13\]
Now on adding \[20\] on both sides to make \[y\] isolated,
We get,
\[\Rightarrow \ - 11y = 13 + 20\]
On simplifying,
We get,
\[\Rightarrow \ - 11y = 33\]
On dividing both sides by \[- 11\] ,
We get,
\[\Rightarrow \ y = \dfrac{33}{- 11}\]
On simplifying,
We get,
\[\Rightarrow \ y = - 3\]
Now on substituting the value of \[y\] in equation (3),
We get,
Equation (3) is \[x = - 4 – 3y\]
On substituting \[y = - 3\] ,
We get,
\[\Rightarrow \ x = - 4 – 3( - 3)\]
On simplifying
We get,
\[\Rightarrow \ x = - 4 + 9\]
On further simplifying,
We get,
\[\Rightarrow \ x = 5\]
Thus we get the solution for the system of equations \[x + 3y = - 4\] and \[5x + 4y = 13\] is \[x = 5\] and \[y = - 3\] .
The solution for the system of equations \[x + 3y = - 4\] and \[5x + 4y = 13\] is \[x = 5\] and \[y = - 3\] .

Note:
To solve these types of questions, we used a substitution method but we can also use other methods like elimination method and augmented matrix method. The most commonly used algebraic methods for solving the linear equations in two variables are the method of elimination by substitution, the method of elimination by equating the coefficients and the method of cross-multiplication. We must know that there is a difference between the methods of solving linear equations in one variable and linear equations in two variables.