Question

Solve by substitution method \[x+2y=-1\] and \[2x-3y=12\]

Answer 1

Hint: In this method of solving equations you should take one of the equations and try to find the value of one variable in terms of another variable. Then put that value of the variable in the second equation so that one of the variables gets eliminated and we can find the value of the other by simply solving a linear equation with one variable.

Complete step-by-step answer:
The given equations are below:
\[\Rightarrow x+2y=-1\] \[...(i)\]
\[\Rightarrow 2x-3y=12\] \[...(ii)\]
First we are going to solve any of the above two equations , let's say equation \[(i)\]and try to find the value of one variable in terms of another.
So now from equation \[(i)\], we get the value of \[x\] in terms of \[y\]
\[\Rightarrow x=-1-2y\] \[...(iii)\]
Now, we will substitute the value of \[x\] we got in equation \[(iii)\]in equation \[(ii)\]so that we can eliminate one of the variables.
And this is called the elimination step in solving linear equations by substitution method.
So now,
Putting \[x's\] value in equation \[(ii)\], we get
\[\Rightarrow 2(-1-2y)-3y=12\]
On solving further,
\[\Rightarrow -2-4y-3y=12\]
\[\Rightarrow -7y=14\]
\[\Rightarrow y=\dfrac{14}{-7}\]
\[\Rightarrow y=-2\]
Here, we got the value of \[y\], now the only step left is to put this value of \[y\] in equation \[(iii)\] to get the value of \[x\].
So now,
Putting \[y's\] value in equation \[(iii)\], we get
\[\Rightarrow x=-1-2y\]
\[\Rightarrow x=-1-2(-2)\]
\[\Rightarrow x=-1+4\]
\[\Rightarrow x=3\]
Here we got the values \[x=3\] and \[y=-2\]
Hence, the solutions of the given equations are \[x=3\], \[y=-2\].
Now we can check further the solution in both original equations and find out the type of the equation is,
Put values of \[x\] and \[y\] in equation \[(i)\]
\[\Rightarrow x+2y=-1\]
\[\Rightarrow 3+2\left( -2 \right)=-1\]
\[\Rightarrow 3-4=-1\]
\[\Rightarrow -1=-1\]
Put values of \[x\] and \[y\] in equation \[(ii)\]
\[\Rightarrow 2x-3y=12\]
\[\Rightarrow 2(3)-3(-2)=12\]
\[\Rightarrow 6+6=12\]
\[\Rightarrow 12=12\]
If the substitution method produces a sentence that is always true, such as \[1=1\] , then the system is dependent, and either original equation is a solution. If the substitution method produces a sentence that is always false, such as \[2=3\], then the system is inconsistent, and there is no solution.
Here we got statements\[12=12\]and \[-1=-1\] that are always true so the given two equations in the question are dependent, i.e. in other words the given equations are dependent on each other.
So, the correct answer is “X = 3 and y = -2”.

Note: These two equations can be solved by other methods also like elimination method , in which we add or subtract two equations in such a way that one variable is cancelled out and we get the value of one variable and put this value in any equation so that we get the value of the second unknown variable.