Let's solve the system of equations step by step to determine which statement is true.
We are given the system of equations:
1. [tex]\( 2y - 4x = 6 \)[/tex]
2. [tex]\( y = 2x + 3 \)[/tex]
#### Step 1: Substitute Equation 2 into Equation 1
Since [tex]\( y = 2x + 3 \)[/tex], we can substitute [tex]\( y \)[/tex] into the first equation:
[tex]\[ 2(2x + 3) - 4x = 6 \][/tex]
#### Step 2: Simplify the Equation
Expand and simplify the left-hand side:
[tex]\[ 4x + 6 - 4x = 6 \][/tex]
This simplifies to:
[tex]\[ 6 = 6 \][/tex]
#### Step 3: Analyze the Result
The equation [tex]\( 6 = 6 \)[/tex] is always true regardless of the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. This indicates that the original equations are not independent; they represent the same line.
#### Finding the Relationship between the Equations
Let's derive [tex]\( y \)[/tex] from both equations to see the relationship more clearly.
From Equation 1:
[tex]\[ 2y - 4x = 6 \][/tex]
[tex]\[ y = 2x + 3 \][/tex]
From Equation 2:
[tex]\[ y = 2x + 3 \][/tex]
Both equations simplify to [tex]\( y = 2x + 3 \)[/tex], which means they are the same line.
#### Conclusion
Since both equations represent the same line, there are infinitely many solutions where any [tex]\((x, y)\)[/tex] that satisfies [tex]\( y = 2x + 3 \)[/tex] is a solution.
#### Answer
Given the analysis above:
D. The system has infinitely many solutions.
