Question

Determine the best method to solve each system of equations. Then solve the system.

$\begin{array}{l}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{12}\\ \mathbf{x}\mathbf{=}\mathbf{-}\mathbf{6}\mathbf{y}\end{array}$

Short answer

The solution of the system of equations is$\mathbf{(}\mathbf{24}\mathbf{,}\mathbf{-}\mathbf{4}\mathbf{)}$.

Step-by-step solution

Step 1. State the concept of substitution method and elimination method.

The substitution method is used to find the set of solutions for the given system of equations. It is the method in which one variable is first solved in the form of another variable.

So, the value of one variable in the form second variable is put on the second equation. It will give the solution of the one variable. And then solve it for another variable.

The elimination method is the method in which addition or subtraction method is used to get an equation in the one variable.

If the coefficients are opposite, use the addition and when the coefficients are same, use the subtraction.

Step 2. Solve at least one equation for one variable and substitute.

The given system of equation is:

$x+3y=12$ … (1)

$x=-6y$ … (2)

The strategy is to solve the system of equations using the best method. Note that in equation (2), variablelocalid="1647580813184" $x$is in terms oflocalid="1647580816823" $y$, therefore, the best method to solve the system is the substitution method.

Now put value oflocalid="1647580820339" $x$from eq. (2) in eq. (1):

$\begin{array}{l}x+3y=12\\ (-6y)+3y=12\text{Substitutex}=-6y\\ -3y=12\\ y=-4\end{array}$

Step 3. Solve for x.

Substitute $-4$for$y$into the equation (2) and solve for$x$.

$\begin{array}{c}x=-6y\\ x=-6(-4)\\ =24\end{array}$

The solution islocalid="1647580838129" $(24,-4)$.