Question

How do you solve this set of equations \[2x+3y=12;\]\[6x+9y=12\]?

Answer 1

Hint: For the first equation the coefficient of x, coefficient of y, and constant term are 2, 3, and 12 respectively. Similarly, for the second equation the coefficient of x, coefficient of y, and constant term are 6, 9, and 12 respectively. If we calculate the values of \[\dfrac{2}{6},\dfrac{3}{9}\And \dfrac{12}{12}\], they are \[\dfrac{1}{3},\dfrac{1}{3}\And 1\]. Thus, we can say that \[\dfrac{2}{6}=\dfrac{3}{9}\ne \dfrac{12}{12}\]. We will use this result later. Now that we are given a system of equations in two variables, we will use the standard steps to solve it and find the solution.

Complete step by step solution:
We are given the two equations \[2x+3y=12\] and \[6x+9y=12\]. We know the steps required to solve a system of equations in two variables. Let’s take the first equation, we get
\[\Rightarrow 2x+3y=12\]
Subtracting \[3y\] from both sides of equation, we get
\[\Rightarrow 2x=12-3y\]
multiplying both sides of the above equation by 3, we get
\[\Rightarrow 6x=36-9y\]
Substituting this in the equation \[6x+9y=12\], we get
\[\begin{align}
  & \Rightarrow 36-9y+9y=12 \\
 & \Rightarrow 36=12 \\
\end{align}\]
But this is never true.

So, this means the given set of linear equations has no solution.

Note: We can also find that the equation has no solution as follows,
Here we can see that, \[\dfrac{a}{m}=\dfrac{b}{n}\ne \dfrac{c}{o}\] here a, b, and c are the x coefficient y coefficient and constant term of first equation. Similarly, m, n, o are the x coefficient y coefficient and constant term of second equation.
If a set of equations is following this property, it means that the set has no solution.