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$\frac{1}{3}x+3y=4$ $\newline$ $2x-4=2y$ $\newline$ Consider the given system of equations. If $(x,y)$ is the solution to the system, then what is the value of $x-y$ ? $\newline$ $\square$

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Full solution

Q.

\frac{1}{3}x+3y=4

\newline

2x-4=2y

\newline

Consider the given system of equations. If

(x,y)

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

x-y

?

\newline

\square

Write Equations: Write down the given system of equations. $\newline$ $(\frac{1}{3})x + 3y = 4$ $\newline$ $2x - 4 = 2y$
Eliminate Fraction: Multiply the first equation by $3$ to eliminate the fraction. $3 \times \left(\frac{1}{3}x + 3y\right) = 3 \times 4$ $x + 9y = 12$
Rearrange Second Equation: Rearrange the second equation to express it in terms of $x$ and $y$ on the same side. $2x - 2y = 4$
New System of Equations: Now we have a system of two equations with two variables: $\newline$ $x + 9y = 12$ $\newline$ $2x - 2y = 4$
Multiply Second Equation: Multiply the second equation by a factor that will allow us to eliminate one of the variables when we add or subtract the equations. In this case, we can multiply the second equation by $\frac{1}{2}$ to make the coefficient of $x$ the same as in the first equation. $\newline$ $\left(\frac{1}{2}\right) \times (2x - 2y) = \left(\frac{1}{2}\right) \times 4$ $\newline$ $x - y = 2$
Final System of Equations: Now we have a new system of two equations: $\newline$ $x + 9y = 12$ $\newline$ $x - y = 2$
Eliminate x: Subtract the second equation from the first to eliminate x. $\newline$ $(x + 9y) - (x - y) = 12 - 2$ $\newline$ $9y + y = 10$ $\newline$ $10y = 10$
Solve for y: Solve for y. $\newline$ y = $\frac{10}{10}$ $\newline$ y = $1$
Substitute for x: Substitute the value of $y$ into the second equation to solve for $x$ . $x - y = 2$ $x - 1 = 2$ $x = 2 + 1$ $x = 3$
Calculate $x - y$ : Calculate the value of $x - y$ . $x - y = 3 - 1$ $x - y = 2$

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