`4x-2y+3=8`
`3x+6y=8y-x+5`
How many solutions does the system of equations shown above have?

Options

A. `0`
B. `1`
C. `2`
D. Indinitely many

Explanation

To determine how many solutions the system of equations has, we can follow these steps:

### Step 1: Write the first equation in standard form.
The first equation is:
\[ 4x - 2y + 3 = 8 \]

To write it in the standard form \( ax + by = c \), we can rearrange it:
\[ 4x - 2y = 8 - 3 \]
\[ 4x - 2y = 5 \]

### Step 2: Write the second equation in standard form.
The second equation is:
\[ 3x + 6y = 8y - x + 5 \]

First, we rearrange it to bring all terms involving \( x \) and \( y \) to one side:
\[ 3x + 6y - 8y + x = 5 \]
Combine like terms:
\[ 4x - 2y = 5 \]

### Step 3: Compare the two equations.
Now we have both equations in standard form:
1. \( 4x - 2y = 5 \) (from the first equation)
2. \( 4x - 2y = 5 \) (from the second equation)

### Step 4: Determine the number of solutions.
Since both equations are identical, they represent the same line on a graph. Therefore, the system of equations has infinitely many solutions.

### Final Answer:
The system of equations has infinitely many solutions.

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