Question

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

$\begin{array}{l}\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{26}\\ \mathbf{2}\mathbf{x}\mathbf{+}\mathbf{3}\mathbf{y}\mathbf{=}\mathbf{19}\end{array}$

Short answer

The best method to solve the system of equations is Elimination method.The solution of the system of equations is $\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{5}\mathbf{)}$.

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. Add or subtract the equations to eliminate one variable.

The given system of equation is:

$3x+4y=26$ … (1)

$2x+3y=19$ … (2)

Multiply equation (1) by 2.

$\begin{array}{l}2(3x+4y)=2\times \left(26\right)\\ 6x+8y=52\text{\hspace{0.17em}.....\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(3\right)\end{array}$

Multiply equation (2) by 3.

$\begin{array}{l}3(2x+3y)=3\times \left(19\right)\\ 6x+9y=57\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.....}\left(4\right)\end{array}$

Note that both the equations (3) and (4) have $6x$in same signs, therefore, variable $x$can be eliminated if both equations are subtracted.

$\begin{array}{l}6x+8y=52\\ \underset{¯}{(-)6x+9y=57}\\ -y=-5\\ y=5\end{array}$

Step 3. Solve for x.

Substitute 5 for y into the equation (1) and solve for $x$.

$\begin{array}{c}3x+4y=26\\ 3x+4\left(5\right)-20=26-20\\ 3x=6\\ x=2\end{array}$

The solution is $(2,5)$.