Question
\( y=2 x+3 \) \( 2 y=4 x+6 \) The system of equations has no solution(s). one infinite
Ask by Fowler Campos. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The system of equations has one infinite solution.
Solution
To determine the nature of the system of equations given by:
1. \( y = 2x + 3 \)
2. \( 2y = 4x + 6 \)
we will analyze the equations step by step.
### Step 1: Simplify the second equation
The second equation can be simplified by dividing all terms by 2:
\[
y = 2x + 3
\]
### Step 2: Compare the equations
Now we have:
1. \( y = 2x + 3 \)
2. \( y = 2x + 3 \)
Both equations are identical. This means that every solution of the first equation is also a solution of the second equation.
### Conclusion
Since both equations represent the same line, the system of equations has infinitely many solutions.
Thus, the answer is that the system of equations has **one infinite solution**.
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Simplify this solution The Deep Dive
Both equations \( y = 2x + 3 \) and \( 2y = 4x + 6 \) represent the same line when simplified. The second equation can be rewritten as \( y = 2x + 3 \) after dividing everything by 2. Since they essentially describe the same relationship, the system of equations has infinitely many solutions, as every point on the line satisfies both equations! To better understand the nature of the system, you can graph these two lines. They'll lie directly on top of one another, demonstrating graphically that an infinite number of solutions exist. This is a perfect example of dependent equations where a single linear equation is represented in two forms!
