The line V passes through the points (-5,3) and (7,-3) and the line W passes through the points (2,-4) and (4,2). The lines V and W intersect at the point A. Work out the coordinates of point A

To find the coordinates of the intersection point \( A \) between the lines \( V \) and \( W \), we need to find the equations of both lines and solve them simultaneously.

### Step 1: Find the equation of line \( V \)
The line \( V \) passes through the points \( (-5, 3) \) and \( (7, -3) \).

First, calculate the slope \( m_V \) of line \( V \) using the formula:
\[
m_V = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 3}{7 - (-5)} = \frac{-6}{12} = -\frac{1}{2}
\]

Now, use the point-slope form of the equation of a line \( y - y_1 = m(x - x_1) \) with one of the points, say \( (-5, 3) \):
\[
y - 3 = -\frac{1}{2}(x - (-5))
\]
\[
y - 3 = -\frac{1}{2}(x + 5)
\]
\[
y - 3 = -\frac{1}{2}x - \frac{5}{2}
\]
\[
y = -\frac{1}{2}x - \frac{5}{2} + \frac{6}{2}
\]
\[
y = -\frac{1}{2}x + \frac{1}{2}
\]

So, the equation of line \( V \) is:
\[
y = -\frac{1}{2}x + \frac{1}{2}
\]

### Step 2: Find the equation of line \( W \)
The line \( W \) passes through the points \( (2, -4) \) and \( (4, 2) \).

First, calculate the slope \( m_W \) of line \( W \) using the formula:
\[
m_W = \frac{2 - (-4)}{4 - 2} = \frac{6}{2} = 3
\]

Now, use the point-slope form with one of the points, say \( (2, -4) \):
\[
y - (-4) = 3(x - 2)
\]
\[
y + 4 = 3(x - 2)
\]
\[
y + 4 = 3x - 6
\]
\[
y = 3x - 6 - 4
\]
\[
y = 3x - 10
\]

So, the equation of line \( W \) is:
\[
y = 3x - 10
\]

### Step 3: Solve the system of equations
Now, solve the two equations simultaneously:
1. \( y = -\frac{1}{2}x + \frac{1}{2} \)
2. \( y = 3x - 10 \)

Set them equal to each other:
\[
-\frac{1}{2}x + \frac{1}{2} = 3x - 10
\]

Multiply through by 2 to eliminate the fractions:
\[
-x + 1 = 6x - 20
\]

Now, solve for \( x \):
\[
1 + 20 = 6x + x
\]
\[
21 = 7x
\]
\[
x = 3
\]

Substitute \( x = 3 \) into one of the original equations, say \( y = 3x - 10 \):
\[
y = 3(3) - 10 = 9 - 10 = -1
\]

### Final Answer:
The coordinates of point \( A \), where the lines intersect, are \( (3, -1) \).

Would you like any further details or clarifications? Here are five related questions you might find interesting:

1. How do you calculate the slope of a line?
2. What is the point-slope form of a line equation?
3. How can you determine if two lines are parallel?
4. What are the conditions for two lines to intersect at right angles?
5. How do you find the distance between two points on a coordinate plane?

**Tip**: When solving simultaneous equations, it is often easier to eliminate fractions by multiplying through by the denominator.

The line V passes through the points (-5,3) and (7,-3) and the line W passes through the points (2,-4) and (4,2). The lines V and W intersect at the point A. Work out the coordinates of point A.

This problem involves finding the intersection point of two lines passing through given points. It requires calculating slopes, writing the equations of the lines, and solving them simultaneously to determine the coordinates of the intersection. Suitable for Grades 9-10, this solution includes step-by-step methods and key concepts like slope and solving simultaneous equations.

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