Question

What is the \[x\] -value in the solution to the system $-6x-5y=10$ & $3x-2y=-6$ ?

Answer 1

Hint: We can easily solve systems of equations like this by simply using the method of substitution. We start solving the problem by expressing the value of \[y\] in terms of \[x\] from the first equation. Then, we consider the expression as a value of \[y\] which we substitute in the other equation. Thus, we get an equation which consists of \[x\] only. After simplifying this equation we get the value of \[x\] .

Complete step-by-step answer:
The equations we are given are $-6x-5y=10$ and $3x-2y=-6$
To solve the set of equations we must find an expression which represents the value of \[y\] i.e., we express \[y\] in terms of the variable \[x\] from one of the equations and substitute that value of \[y\] in the other equation, solving which we obtain the value of \[x\] .
Now we start solving the problem by taking the first equation as
$-6x-5y=10$
We add \[5y\] to both the sides of the equation as shown below
$\Rightarrow -6x-5y+5y=10+5y$
$\Rightarrow -6x=10+5y$
Again, we subtract \[10\] from both the sides of the equation as shown below
$\Rightarrow -6x-10=10+5y-10$
$\Rightarrow -6x-10=5y$
We can rewrite the above expression as
$\Rightarrow y=\dfrac{-6x-10}{5}$
The above expression is the value of \[y\] which is represented in terms of \[x\] . We now substitute this value in the other equation as shown below
$\Rightarrow 3x-2\left( \dfrac{-6x-10}{5} \right)=-6$
Further simplifying the above equation, we get
$\Rightarrow 15x+12x+20=-30$
$\Rightarrow x=-\dfrac{50}{27}$
Therefore, the required given is $x=-\dfrac{50}{27}$ .

Note: Instead of using the substitution method we can also use another method, which is the elimination method. In the elimination method we add or subtract the two equations as per our requirements to eliminate one of the variables which gives the value of the other variable. Putting this value in any of the given equations will give us the value of the other variable.