batguy2
Answered

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This system of equations is shown on the graph:
[tex]\[
\begin{array}{l}
2y - 4x = 6 \\
y = 2x + 3
\end{array}
\][/tex]

Which statement about the system is true?

A. The system has no solution.
B. The system has a unique solution at [tex]\((0, 3)\)[/tex].
C. The system has a unique solution at [tex]\((1, 5)\)[/tex].
D. The system has infinitely many solutions.

Sagot :

GinnyAnswer
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.
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