Full solution

3 x+4(2-y)=5 x

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2-y=6 x

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Consider the given system of equations. If

(x, y)

is the solution to the system, then what is the value of

\frac{x}{y}

Solve for y: Solve the first equation for y. $\newline$ We have the equation $3x + 4(2 - y) = 5x$ . $\newline$ First, distribute the $4$ into the parentheses: $3x + 8 - 4y = 5x$ . $\newline$ Now, isolate the $y$ term by moving the other terms to the other side: $-4y = 5x - 3x - 8$ . $\newline$ Simplify the right side: $-4y = 2x - 8$ . $\newline$ Divide both sides by $-4$ to solve for $y$ : $y = (2x - 8) / -4$ . $\newline$ Simplify the right side: $y = -0.5x + 2$ .
Substitute $y$ into the second equation: Substitute the expression for $y$ into the second equation. $\newline$ We have the second equation $2 - y = 6x$ . $\newline$ Substitute $y$ with the expression found in Step $1$ : $2 - (-0.5x + 2) = 6x$ . $\newline$ Simplify the equation: $2 + 0.5x - 2 = 6x$ . $\newline$ The $2$ 's on the left side cancel out, leaving us with: $0.5x = 6x$ .
Solve for x: Solve for x. $\newline$ We have the equation $0.5x = 6x$ . $\newline$ Subtract $0.5x$ from both sides to get all $x$ terms on one side: $0.5x - 0.5x = 6x - 0.5x$ . $\newline$ This simplifies to $0 = 5.5x$ . $\newline$ Divide both sides by $5.5$ to solve for $x$ : $0 / 5.5 = x$ . $\newline$ This gives us $x = 0$ .
Substitute $x$ into the expression for $y$ : Substitute $x = 0$ into the expression for $y$ . We have $y = -0.5x + 2$ . $\newline$ Substitute $x$ with $0$ : $y = -0.5(0) + 2$ . $\newline$ Simplify the equation: $y = 0 + 2$ . $\newline$ This gives us $y = 2$ .
Calculate $\frac{x}{y}$ : Calculate the value of $\frac{x}{y}$ . $\newline$ We have $x = 0$ and $y = 2$ . $\newline$ Substitute these values into the expression: $\frac{x}{y} = \frac{0}{2}$ . $\newline$ Simplify the expression: $\frac{x}{y} = 0$ .