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How many solutions does the system have? $\newline$ $\begin{cases} x+y=3 \\ \ 5x+5y=15 \end{cases}$

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Estimate customary measurements

Full solution

Q. How many solutions does the system have?

\newline

\begin{cases} x+y=3 \\ \ 5x+5y=15 \end{cases}

Write Equations: First, let's write down the system of equations to analyze it. $\begin{cases} x + y = 3 \ 5x + 5y = 15 \end{cases}$
Observe Equation Relationship: Next, we observe that the second equation is simply the first equation multiplied by $5$ . This means that the two equations are actually the same line represented in different forms.
Identify Infinite Solutions: Since both equations represent the same line, every point on the line is a solution to both equations. Therefore, the system does not have a unique solution; instead, it has infinitely many solutions.
Confirm Observation: To confirm our observation, we can divide the second equation by $5$ to see if it simplifies to the first equation. $\newline$ $\frac{5x + 5y}{5} = \frac{15}{5}$ $\newline$ $x + y = 3$ $\newline$ This confirms that the second equation is indeed a multiple of the first.

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