Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution. $\newline$ $\begin{array}{c} 4 x+y=4 \\ -6 x-y=-6 \end{array}$ $\newline$ One Solution $\newline$ Infinitely Many Solutions $\newline$ No Solutions

Full solution

Q. Determine if the following system of equations has no solutions, infinitely many solutions or exactly one solution.

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\begin{array}{c} 4 x+y=4 \\ -6 x-y=-6 \end{array}

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One Solution

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Infinitely Many Solutions

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No Solutions

Write Equations: Write down the system of equations to analyze. $\begin{cases} 4x+y=4 \ -6x-y=-6 \end{cases}$
Add and Eliminate $y$ : Attempt to solve the system by adding the two equations together to eliminate $y$ . $\newline$ Adding $4x + y = 4$ and $-6x - y = -6$ gives us: $\newline$ $4x + y - 6x - y = 4 - 6$ $\newline$ This simplifies to: $\newline$ $-2x = -2$
Solve for x: Solve for x by dividing both sides of the equation by \(-2＂). $\newline$ \(-2x / $-2$ = $-2$ / $-2$ ＂) $\newline$ This gives us: $\newline$ x = \(1＂)
Substitute $x$ into Equation: Substitute $x = 1$ into one of the original equations to solve for $y$ . Using the first equation $4x + y = 4$ : $4(1) + y = 4$ $4 + y = 4$ Subtract $4$ from both sides to solve for $y$ : $y = 4 - 4$ $y = 0$
Check Validity: Check the solution $(x = 1, y = 0)$ in the second equation to ensure it is valid. $\newline$ Substitute $x = 1$ and $y = 0$ into $-6x - y = -6$ : $\newline$ $-6(1) - 0 = -6$ $\newline$ $-6 = -6$ $\newline$ The solution satisfies the second equation as well.
Final Solution: Since we have found a single solution $(x = 1, y = 0)$ that satisfies both equations, the system has exactly one solution.