The problem asks us to perform several tasks. First, we need to find the intersection points of pairs of lines given by their equations. Specifically, we have to solve for the intersection of lines in the following pairs: 1) $x - 2y - 4 = 0$ and $2x + 3y - 1 = 0$ 2) $x + 5y - 15 = 0$ and $2x - y + 3 = 0$ 3) $x - 2y - 1 = 0$ and $3x + y - 3 = 0$ 4) $2x + y - 6 = 0$ and $3x - y - 4 = 0$

1. Problem Description

The problem asks us to perform several tasks. First, we need to find the intersection points of pairs of lines given by their equations. Specifically, we have to solve for the intersection of lines in the following pairs:

1) $x - 2y - 4 = 0$ and $2x + 3y - 1 = 0$

2) $x + 5y - 15 = 0$ and $2x - y + 3 = 0$

3) $x - 2y - 1 = 0$ and $3x + y - 3 = 0$

4) $2x + y - 6 = 0$ and $3x - y - 4 = 0$

2. Solution Steps

We solve each pair of equations by substitution or elimination.

1) $x - 2y - 4 = 0$ and $2x + 3y - 1 = 0$

From the first equation, $x = 2y + 4$. Substituting into the second equation:

$2(2y + 4) + 3y - 1 = 0$

$4y + 8 + 3y - 1 = 0$

$7y + 7 = 0$

$7y = -7$

$y = -1$

$x = 2(-1) + 4 = -2 + 4 = 2$

Intersection point: $(2, -1)$

2) $x + 5y - 15 = 0$ and $2x - y + 3 = 0$

From the first equation, $x = 15 - 5y$. Substituting into the second equation:

$2(15 - 5y) - y + 3 = 0$

$30 - 10y - y + 3 = 0$

$33 - 11y = 0$

$11y = 33$

$y = 3$

$x = 15 - 5(3) = 15 - 15 = 0$

Intersection point: $(0, 3)$

3) $x - 2y - 1 = 0$ and $3x + y - 3 = 0$

From the first equation, $x = 2y + 1$. Substituting into the second equation:

$3(2y + 1) + y - 3 = 0$

$6y + 3 + y - 3 = 0$

$7y = 0$

$y = 0$

$x = 2(0) + 1 = 1$

Intersection point: $(1, 0)$

4) $2x + y - 6 = 0$ and $3x - y - 4 = 0$

Add the two equations:

$(2x + y - 6) + (3x - y - 4) = 0$

$5x - 10 = 0$

$5x = 10$

$x = 2$

Substitute $x = 2$ into the first equation:

$2(2) + y - 6 = 0$

$4 + y - 6 = 0$

$y - 2 = 0$

$y = 2$

Intersection point: $(2, 2)$

3. Final Answer

1) Intersection point: $(2, -1)$

2) Intersection point: $(0, 3)$

3) Intersection point: $(1, 0)$

4) Intersection point: $(2, 2)$