Question

solve the system by either the addition method or the substitution method.\

$$\begin{cases}4y = x + 8\\\\5x - 4y = -8\\end{cases}$$

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

Explanation:

Step1: Use addition method

We have the system:

$$\begin{cases} 4y = x + 8 \\ 5x - 4y = -8 \end{cases}$$

Rewrite the first equation as $-x + 4y = 8$. Now we have:

$$\begin{cases} -x + 4y = 8 \\ 5x - 4y = -8 \end{cases}$$

Step2: Add the two equations

Add the left sides and the right sides of the two equations:
$$(-x + 4y)+(5x - 4y)=8+(-8)$$
Simplify the left side: $-x + 5x+4y - 4y = 4x$
Simplify the right side: $8 - 8=0$
So we get $4x = 0$, which implies $x = 0$.

Step3: Substitute x = 0 into the first equation

Substitute $x = 0$ into $4y=x + 8$:
$$4y=0 + 8$$
Simplify: $4y = 8$, then $y=\frac{8}{4}=2$

Answer:

A. The solution is \((0, 2)\)