Question

solve the system of equations by the addition method. \

$$\begin{cases} 3x + y = 19 \\\\ 4x - y = 16 \\end{cases}$$

select the correct choice below and, if necessary, fill in the answer box to complete your choice. \
\
a. the solution is \boxed{}. (simplify your answer. type an ordered pair.) \
b. there are infinitely many solutions. \
c. there is no solution.

Explanation:

Step1: Add the two equations

We have the system:

$$\begin{cases} 3x + y = 19 \\ 4x - y = 16 \end{cases}$$

Adding the two equations together: $(3x + y)+(4x - y)=19 + 16$.
Simplifying the left - hand side: $3x+y + 4x-y=7x$, and the right - hand side is $35$. So we get the equation $7x=35$.

Step2: Solve for x

To solve for $x$, divide both sides of the equation $7x = 35$ by $7$: $x=\frac{35}{7}=5$.

Step3: Substitute x into one of the original equations

Substitute $x = 5$ into the first equation $3x + y=19$. We have $3\times5+y = 19$, which simplifies to $15 + y=19$.

Step4: Solve for y

Subtract $15$ from both sides of the equation $15 + y=19$: $y=19 - 15=4$.

Answer:

A. The solution is \((5,4)\)