Problems from Method of equalization

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: $$$3x-2(3y+5)-10=7 \Rightarrow 3x-6y-10-10=7 \Rightarrow 3x-6y=7+10+10 \Rightarrow$$$ $$$\Rightarrow 3x-6y=27$$$ Also, we can divide all the coefficients by $$3$$, so that we obtain $$ \dfrac{3x-6y=27}{3} \Rightarrow x-2y=9$$.

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

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

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

Solution:

$$x=3; y=-3$$

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Development:

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

$$\left.\begin{array}{c} x+\dfrac{y}{3}=3 \\ \dfrac{x}{2}-y=-2 \end{array} \right\}$$

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

$$\left.\begin{array}{c} 3\cdot\Big[x+\dfrac{y}{3}=3\Big] \\ 2\cdot\Big[\dfrac{x}{2}-y=-2\Big] \end{array} \right\} \Rightarrow \left.\begin{array}{c} 3x+y=9 \\ x-2y=-4 \end{array} \right\} $$

This system is completely equivalent to the first one.

We can express x in the first equation in terms of $$y$$: $$$3x=9-y \Rightarrow x=\dfrac{9-y}{3}=3-\dfrac{y}{3}$$$ And do the same in the second equation: $$$x=2y-4$$$ Now both expressions have to be equal and we know how to solve for the resulting equation: $$$3-\dfrac{y}{3}=2y-4 \Rightarrow -\dfrac{y}{3}-2y=-4-3 \Rightarrow \dfrac{-y-6y}{3}=-7 \Rightarrow -7y=-21 \Rightarrow$$$ $$$y=\dfrac{-21}{-7}=3$$$ 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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