Question

How do you solve: $x + y = 20$ and $x = 3y$ ?

Answer 1

Hint: In the given question, we need to solve two simultaneous equations in two variables. There are various methods to solve two given equations in two variables like substitution method, cross multiplication method, elimination method, matrix method and many more. The equations given in the question can be solved using any one of the above mentioned methods easily. But we will solve the equations using the substitution method. We will substitute the value of one variable in terms of another into the second equation.

Complete step by step answer:
In the question, we are given a couple of simultaneous linear equations in two variables.
$x + y = 20 - - - - - \left( 1 \right)$
$\Rightarrow x = 3y - - - - \left( 2 \right)$
In the substitution method, we substitute the value of one variable from an equation into another equation so as to get an equation in only one variable. So, from equation $\left( 1 \right)$, we get,
$ \Rightarrow x = 20 - y - - - - \left( 3 \right)$

Now putting the value of x obtained from one equation into another. So, we equate the values of x from the equations $\left( 2 \right)$ and $\left( 3 \right)$.
So, we get, $3y = 20 - y$
Shifting all the terms consisting y to the left side of the equation and all the constants to the right side of the equation, we get,
$ \Rightarrow 3y + y = 20$
Simplifying the calculations, we get,
$ \Rightarrow 4y = 20$

Dividing both sides of equation by four, we get,
$ \Rightarrow y = \dfrac{{20}}{4}$
$ \Rightarrow y = 5$
So, we get the value of y as five.
Now, putting the value of y in any of the two equations, we get,
$x = 3y$
$ \Rightarrow x = 3\left( 5 \right)$
Carrying out the multiplication, we get,
$ \Rightarrow x = 15$

Therefore,the solution of the simultaneous linear equations $x + y = 20$ and $x = 3y$ is $x = 15$ and $y = 5$.

Note: An equation consisting of two variables having degree one is known as Linear Equation in two variables. Standard form of Linear Equation in two variables is $ax + by + c = 0$ where a, b and c are the real numbers and a, b which are coefficients of x and y respectively are not equal to zero.