Question

two systems of equations are given below.
for each system, choose the best description of its solution.
if applicable, give the solution.

(a) solve the following system of equations.
\\\

$$\begin{cases} x + 4y = 8 \\\\ -x - 4y = 8 \\end{cases}$$

\\

• the system has no solution.
• the system has a unique solution.

\\((x, y) = (\square, \square)\\)

• the system has infinitely many solutions. they must satisfy the following equation.

\\(y = \square\\)

(b) solve the following system of equations.
\\\

$$\begin{cases} -x - 2y = -2 \\\\ x + 2y = 2 \\end{cases}$$

\\

• the system has no solution.
• the system has a unique solution.

\\((x, y) = (\square, \square)\\)

Explanation:

Analyze system (a)

$$
LATEXBLOCK0
$$

$$ (x + 4y) + (-x - 4y) = 8 + 8 \implies 0 = 16 $$

Analyze system (b)

$$
LATEXBLOCK1
$$

$$ (-x - 2y) + (x + 2y) = -2 + 2 \implies 0 = 0 $$

$$ x + 2y = 2 \implies 2y = -x + 2 \implies y = -\frac{1}{2}x + 1 $$

Answer:

Question 1

The system has no solution.

Question 2

The system has infinitely many solutions. They must satisfy the following equation.
\(y = -\frac{1}{2}x + 1\)