Question

1. ( y = \frac{1}{3}x + 3 )

( y = \frac{1}{3}x - 2 )

1. ( y = -\frac{5}{2}x - 2 )

( y = \frac{1}{2}x + 4 )

Explanation:

Step1: Analyze the equations

We have two linear equations: \( y = -\frac{5}{2}x - 2 \) and \( y=\frac{1}{2}x + 4 \). To find their intersection (solution), we set them equal:
\( -\frac{5}{2}x - 2=\frac{1}{2}x + 4 \)

Step2: Solve for x

Add \( \frac{5}{2}x \) to both sides:
\( - 2=\frac{1}{2}x+\frac{5}{2}x + 4 \)
Combine like terms: \( \frac{1 + 5}{2}x=\frac{6}{2}x = 3x \), so \( -2 = 3x+4 \)
Subtract 4 from both sides: \( -2-4=3x \) → \( -6 = 3x \)
Divide by 3: \( x=\frac{-6}{3}=-2 \)

Step3: Solve for y

Substitute \( x = - 2 \) into \( y=\frac{1}{2}x + 4 \):
\( y=\frac{1}{2}(-2)+4=-1 + 4 = 3 \)

Answer:

The solution to the system \(

$$\begin{cases}y = -\frac{5}{2}x - 2\\y=\frac{1}{2}x + 4\end{cases}$$

\) is \( x=-2,y = 3 \) (the point \((-2,3)\)).