Let's solve the system of linear equations step-by-step to determine the correct statement.
The given system of equations is:
1. [tex]\(2y - 4x = 6\)[/tex]
2. [tex]\(y = 2x + 3\)[/tex]
To find the solution, we can substitute the expression for [tex]\(y\)[/tex] from the second equation into the first equation.
### Step 1: Substitute [tex]\(y = 2x + 3\)[/tex] into [tex]\(2y - 4x = 6\)[/tex]
The first equation is:
[tex]\[2y - 4x = 6\][/tex]
Plug in [tex]\(y = 2x + 3\)[/tex]:
[tex]\[2(2x + 3) - 4x = 6\][/tex]
### Step 2: Simplify the equation
Distribute the 2:
[tex]\[4x + 6 - 4x = 6\][/tex]
### Step 3: Combine like terms
[tex]\[4x - 4x + 6 = 6\][/tex]
[tex]\[0x + 6 = 6\][/tex]
[tex]\[6 = 6\][/tex]
### Analysis:
The equation simplifies to [tex]\(6 = 6\)[/tex], which is always true regardless of the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex]. This means the two equations are not strict lines. Instead, the first equation is just a multiple of the second.
### Conclusion:
When you substitute [tex]\(y = 2x + 3\)[/tex] into [tex]\(2y - 4x = 6\)[/tex] and arrive at a tautology (a true statement like [tex]\(6 = 6\)[/tex]), it indicates that the two lines are actually the same line.
Therefore, the equations represent the same line, and there are infinitely many solutions since every point on the line [tex]\(y = 2x + 3\)[/tex] is a solution to the system.
The correct answer is:
D. The system has infinitely many solutions.
