Problems from Method of equalization

Solve the following system with the equalization method:

$\begin{matrix} 3 x - 2 (3 y + 5) - 10 = 7 \\ 4 (x - 3) + 2 y = - 3 + y \end{matrix}}$

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Development:

Before using the equalization method, we need to simplify the equations a little bit. We will obtain a simpler system of equations with the same solution. In the first equation: $3 x - 2 (3 y + 5) - 10 = 7 \Rightarrow 3 x - 6 y - 10 - 10 = 7 \Rightarrow 3 x - 6 y = 7 + 10 + 10 \Rightarrow$ $\Rightarrow 3 x - 6 y = 27$ Also, we can divide all the coefficients by $3$ , so that we obtain $\frac{3 x - 6 y = 27}{3} \Rightarrow x - 2 y = 9$ .

The second equation of the system is simplified as follows: $4 (x - 3) + 2 y = - 3 + y \Rightarrow 4 x - 12 + 2 y - y = - 3 \Rightarrow 4 x + y = - 3 + 12 \Rightarrow 4 x + y = 9$ With these two simplified equations we can use the equalization method: $\begin{matrix} x - 2 y = 9 \\ 4 x + y = 9 \end{matrix}} \Rightarrow \begin{matrix} x = 9 + 2 y \\ 4 x = 9 - y \end{matrix}} \Rightarrow \begin{matrix} x = 9 + 2 y \\ x = \frac{9 - y}{4} \end{matrix}}$

The expressions need to be equal and we have $y$ : $9 + 2 y = \frac{9 - y}{4} \Rightarrow 4 (9 + 2 y) = 9 - y \Rightarrow 36 + 8 y = 9 - y \Rightarrow 8 y + y = 9 - 36 \Rightarrow$ $9 y = - 27 \Rightarrow y = \frac{- 27}{9} = - 3$

The value is then replaced in order to obtain $x$ : $x = 9 + 2 y \Rightarrow x = 9 + 2 (- 3) = 9 - 6 = 3$

Solution:

$x = 3; y = - 3$

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Solve the following system with the equalization method:

$\begin{matrix} x - 1 = 2 - \frac{y}{3} \\ 1 - y = - 1 - \frac{x}{2} \end{matrix}}$

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Development:

We can start to put all the variables together on one side and the constants onto the other:

$\begin{matrix} x + \frac{y}{3} = 3 \\ \frac{x}{2} - y = - 2 \end{matrix}}$

Now we can eliminate the fractions. We have to multiply the first equation by $3$ and the second one by $2$ :

$\begin{matrix} 3 \cdot [x + \frac{y}{3} = 3] \\ 2 \cdot [\frac{x}{2} - y = - 2] \end{matrix}} \Rightarrow \begin{matrix} 3 x + y = 9 \\ x - 2 y = - 4 \end{matrix}}$

This system is completely equivalent to the first one.

We can express x in the first equation in terms of $y$ : $3 x = 9 - y \Rightarrow x = \frac{9 - y}{3} = 3 - \frac{y}{3}$ And do the same in the second equation: $x = 2 y - 4$ Now both expressions have to be equal and we know how to solve for the resulting equation: $3 - \frac{y}{3} = 2 y - 4 \Rightarrow - \frac{y}{3} - 2 y = - 4 - 3 \Rightarrow \frac{- y - 6 y}{3} = - 7 \Rightarrow - 7 y = - 21 \Rightarrow$ $y = \frac{- 21}{- 7} = 3$ The value is then replaced in order to obtain $x$ : $x = 2 \cdot (3) - 4 = 6 - 4 = 2$

Solution:

$x = 2; y = 3$

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