Analyzing Straight Line Equations

We are given three straight lines:

• Line 1: $x + y - 3 = 0$
• Line 2: $x + y + 2 = 0$
• Line 3: $3x + 3y - 7 = 0$

To determine their relationship, we need to find the slope of each line. The slope ($m$) of a line in the form $Ax + By + C = 0$ is given by $m = -A/B$.

Calculating Slopes

• For Line 1 ($x + y - 3 = 0$): $A=1, B=1$. Slope $m_1 = -1/1 = -1$.
• For Line 2 ($x + y + 2 = 0$): $A=1, B=1$. Slope $m_2 = -1/1 = -1$.
• For Line 3 ($3x + 3y - 7 = 0$): $A=3, B=3$. Slope $m_3 = -3/3 = -1$.

Determining Line Relationship

We observe that the slopes of all three lines are equal ($m_1 = m_2 = m_3 = -1$).

Lines with equal slopes are either parallel or the same line.

To check if they are the same line, we compare their constant terms (C):

• Line 1: $C_1 = -3$
• Line 2: $C_2 = 2$
• Line 3: $C_3 = -7$

Since the constant terms are different, the lines are distinct.

Three distinct lines with the same slope are parallel to each other.