Find the joint equation of lines y =x and y=-x.

Answer

To find the joint equation of the lines \( y = x \) and \( y = -x \), we can follow these steps:

### Step 1: Convert the equations of the lines to standard form
The equations of the lines are given as:
1. \( y = x \)
2. \( y = -x \)

We can rewrite these equations in the standard form \( Ax + By + C = 0 \).

For the first line \( y = x \):
\[ y - x = 0 \quad \text{(or)} \quad -x + y + 0 = 0 \]
This gives us \( A_1 = -1, B_1 = 1, C_1 = 0 \).

For the second line \( y = -x \):
\[ y + x = 0 \quad \text{(or)} \quad x + y + 0 = 0 \]
This gives us \( A_2 = 1, B_2 = 1, C_2 = 0 \).

### Step 2: Write the joint equation
The joint equation of two lines can be found using the formula:
\[ (A_1 x + B_1 y + C_1)(A_2 x + B_2 y + C_2) = 0 \]

Substituting the values we found:
\[ (-x + y + 0)(x + y + 0) = 0 \]

### Step 3: Expand the joint equation
Now we will expand the expression:
\[ (-x + y)(x + y) = 0 \]

Using the distributive property:
\[ -x^2 - xy + yx + y^2 = 0 \]
The \( -xy \) and \( yx \) cancel each other out:
\[ -x^2 + y^2 = 0 \]

### Step 4: Rearranging the equation
We can rearrange this to:
\[ y^2 - x^2 = 0 \]
This can also be expressed as:
\[ x^2 - y^2 = 0 \]

### Final Answer
Thus, the joint equation of the lines \( y = x \) and \( y = -x \) is:
\[ x^2 - y^2 = 0 \]