$3х + 2y = 4(x-y-6)$ $\newline$ $6(у + x)=7x-24$ $\newline$ Which of the following accurately describes all solutions to the system of equations shown?

Full solution

3х + 2y = 4(x-y-6)

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6(у + x)=7x-24

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Which of the following accurately describes all solutions to the system of equations shown?

Expand and Simplify Equations: Expand the equations to simplify them. $\newline$ For the first equation, distribute the $4$ to each term inside the parentheses: $\newline$ $3x + 2y = 4(x - y - 6)$ $\newline$ $3x + 2y = 4x - 4y - 24$
Rearrange and Isolate Variables: Rearrange the first equation to group like terms and isolate variables. $\newline$ Subtract $3x$ from both sides and add $4y$ to both sides: $\newline$ $3x + 2y - 3x + 4y = 4x - 4y - 24 - 3x + 4y$ $\newline$ $2y + 4y = 4x - 3x - 24$ $\newline$ $6y = x - 24$
Simplify Second Equation: Simplify the second equation by distributing the $6$ to each term inside the parentheses: $6(y + x) = 7x - 24$ $6y + 6x = 7x - 24$
Rearrange and Isolate Variables: Rearrange the second equation to group like terms and isolate variables. $\newline$ Subtract $6x$ from both sides: $\newline$ $6y + 6x - 6x = 7x - 24 - 6x$ $\newline$ $6y = x - 24$
Identical Equations Observation: Observe that both simplified equations are identical. $\newline$ $6y = x - 24$ (from Step $2$ ) $\newline$ $6y = x - 24$ (from Step $4$ ) $\newline$ This means that the two equations are dependent and represent the same line.
Conclusion of Infinite Solutions: Conclude that the system of equations has infinitely many solutions because both equations represent the same line.