Question

solve the system by the addition method.\

$$\begin{cases}x + 2y = 3\\\\2x + 5y = 8\\end{cases}$$

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

Explanation:

Step1: Eliminate x by multiplying the first equation

Multiply the first equation \(x + 2y = 3\) by \(-2\) to get \(-2x - 4y = -6\).

Step2: Add the two equations

Add the new equation \(-2x - 4y = -6\) to the second equation \(2x + 5y = 8\):
\[

$$\begin{align*} (-2x - 4y) + (2x + 5y) &= -6 + 8\\ -2x - 4y + 2x + 5y &= 2\\ y &= 2 \end{align*}$$

\]

Step3: Substitute y into the first equation

Substitute \(y = 2\) into \(x + 2y = 3\):
\[

$$\begin{align*} x + 2(2) &= 3\\ x + 4 &= 3\\ x &= 3 - 4\\ x &= -1 \end{align*}$$

\]

Answer:

A. The solution is \((-1, 2)\)