Question

classify each system. 5. \

$$\begin{cases} y + 2(x - 3) = 0 \\\\ 2x = -y - 3 \\end{cases}$$

Explanation:

Step1: Simplify the first equation

Expand and rearrange \( y + 2(x - 3)=0 \):
\( y + 2x - 6 = 0 \), so \( 2x + y = 6 \).

Step2: Rewrite the second equation

Rearrange \( 2x = -y - 3 \):
\( 2x + y = -3 \).

Step3: Analyze the two equations

The two equations are \( 2x + y = 6 \) and \( 2x + y = -3 \). These are two linear equations in two variables. The left - hand sides are the same (\( 2x + y \)), but the right - hand sides are different (\( 6\) and \(-3\)). For a system of linear equations \( a_1x + b_1y = c_1 \) and \( a_2x + b_2y = c_2 \), if \( \frac{a_1}{a_2}=\frac{b_1}{b_2}
eq\frac{c_1}{c_2} \) (here \( a_1 = 2,a_2 = 2,b_1 = 1,b_2 = 1,c_1 = 6,c_2=-3 \), \( \frac{2}{2}=\frac{1}{1}
eq\frac{6}{-3} \)), the system is inconsistent (has no solution) and the lines are parallel.

Answer:

The system of equations \(

$$\begin{cases}y + 2(x - 3)=0\\2x=-y - 3\end{cases}$$

\) is an inconsistent system of linear equations (it has no solution, and the two lines represented by the equations are parallel).