Find the number of solution of the system of equation x+2y=6 and |x-3|=y

Explanation

To find the number of solutions for the system of equations given by:

1. \( x + 2y = 6 \)
2. \( |x - 3| = y \)

we will analyze the second equation, which involves the absolute value.

### Step 1: Analyze the absolute value equation

The equation \( |x - 3| = y \) can be split into two cases based on the definition of absolute value:

- Case 1: \( x - 3 = y \) when \( x \geq 3 \)
- Case 2: \( 3 - x = y \) when \( x < 3 \)

### Step 2: Solve Case 1

For Case 1, substitute \( y = x - 3 \) into the first equation:

\[ x + 2(x - 3) = 6 \]

Simplifying this:

\[ x + 2x - 6 = 6 \]
\[ 3x - 6 = 6 \]
\[ 3x = 12 \]
\[ x = 4 \]

Now, since \( x = 4 \) is greater than or equal to 3, we can find \( y \):

\[ y = 4 - 3 = 1 \]

Thus, one solution is \( (4, 1) \).

### Step 3: Solve Case 2

For Case 2, substitute \( y = 3 - x \) into the first equation:

\[ x + 2(3 - x) = 6 \]

Simplifying this:

\[ x + 6 - 2x = 6 \]
\[ -x + 6 = 6 \]
\[ -x = 0 \]
\[ x = 0 \]

Now, since \( x = 0 \) is less than 3, we can find \( y \):

\[ y = 3 - 0 = 3 \]

Thus, another solution is \( (0, 3) \).

### Step 4: Conclusion

We have found two solutions for the system of equations:

1. \( (4, 1) \)
2. \( (0, 3) \)

Therefore, the number of solutions to the system of equations is **2**.

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