Question
\( \begin{array}{l}y=2 x+3 \\ 2 y=4 x+6\end{array} \) The system of equations has \( \quad \) infinite one no
Ask by Nguyen Bernard. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The system has infinitely many solutions.
Solution
We are given the system of equations:
1) y = 2x + 3
2) 2y = 4x + 6
Step 1: Simplify Equation 2.
Divide both sides of Equation 2 by 2:
y = (4x + 6) / 2
y = 2x + 3
Step 2: Compare with Equation 1.
Notice that Equation 2 simplifies exactly to Equation 1. This means both equations represent the same line.
Conclusion:
Since both equations are identical, every point on the line y = 2x + 3 is a solution. Thus, the system has infinitely many solutions.
Answer: infinite
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Simplify this solution The Deep Dive
The equations \(y = 2x + 3\) and \(2y = 4x + 6\) are actually the same line. If you manipulate the second equation by dividing everything by 2, you’ll see it simplifies to \(y = 2x + 3\). This means that there are infinitely many solutions since every point on the line is a solution! In the realm of linear equations, when two equations represent the same line, they are considered dependent, leading to infinite solutions. To visualize this, you could graph both equations on the same set of axes—surprise! They overlap completely, confirming they are one and the same!
