Elige el método más adecuado para resolver los siguientes sistemas de ecuaciones: a. \begin{cases} x - 4y + 5 = 2y + 3x - 3 \\ 6x = 1 + 5y \end{cases} b. \begin{cases} 2.8x - 1.7... Elige el método más adecuado para resolver los siguientes sistemas de ecuaciones: a. \begin{cases} x - 4y + 5 = 2y + 3x - 3 \\ 6x = 1 + 5y \end{cases} b. \begin{cases} 2.8x - 1.7y = 8.9 \\ 3.2(x - 3y) = -12.8 \end{cases} c. \begin{cases} 2x + \frac{y}{4} = 5 \\ -3(x - 1) + 13 = 2(y + 1) \end{cases} d. \begin{cases} 2(5x - 3) + 2y = 0 \\ -3(x - y) = -(x + 8) \end{cases} e. \begin{cases} 2x - y + 3(x - y) = -4 \\ 5(x - 3y) + 2y - 3(3x - 2y) = -7 \end{cases} Read more

Steps to Solve

a. Simplifying the equations

First, simplify both equations to put them in a standard form.

Equation 1: $x - 4y + 5 = 2y + 3x - 3$ simplifies to $-2x - 6y = -8$, or $x + 3y = 4$.

Equation 2: $6x = 1 + 5y$ can be rewritten as $6x - 5y = 1$.

Choosing the method

Since we can easily isolate $x$ in the first equation ($x = 4 - 3y$), substitution is a good choice.

b. Simplifying the equations

Simplify the second equation: $3.2(x - 3y) = -12.8$ becomes $3.2x - 9.6y = -12.8$.

Choosing the method

Notice that we could divide the second equation by $3.2$ to get $x-3y=-4$ Then isolating $x$ gives is $x = 3y - 4$. Given the decimals in the first equation, isolating $x$ or $y$ and then using substitution seems like the easiest choice. Substitution would be ideal here.

c. Simplifying the equations

Simplify both equations.

Equation 1: $2x + \frac{y}{4} = 5$

Equation 2: $-3(x - 1) + 13 = 2(y + 1)$ becomes $-3x + 3 + 13 = 2y + 2$, simplifying to $-3x - 2y = -14$.

Choosing the method

Clearing the fraction in the first equation results in $8x + y = 20$. Isolating $y$ in the first equation is simple: $y = 20 - 8x$. So substitution would work Alternatively, multiplying the first equation by 8 and the bottom by 1 may make elimination easier.

d. Simplifying the equations

Simplify both equations.

Equation 1: $2(5x - 3) + 2y = 0$ becomes $10x - 6 + 2y = 0$, or $10x + 2y = 6$, which simplifies to $5x + y = 3$.

Equation 2: $-3(x - y) = -(x + 8)$ becomes $-3x + 3y = -x - 8$, simplifying to $-2x + 3y = -8$.

Choosing the method

From the first equation, we get $y = 3 - 5x$. Then we can plug this into the second equation. Alternatively, we could multiply the first equation by $-3$ and use elimination. Substitution seems like our best bet.

e. Simplifying the equations

Simplify both equations.

Equation 1: $2x - y + 3(x - y) = -4$ becomes $2x - y + 3x - 3y = -4$, simplifying to $5x - 4y = -4$.

Equation 2: $5(x - 3y) + 2y - 3(3x - 2y) = -7$ becomes $5x - 15y + 2y - 9x + 6y = -7$, simplifying to $-4x - 7y = -7$.

Choosing the method

We have $5x - 4y = -4$ and $-4x - 7y = -7$.

We do not have any easy options for substitution here. Therefore, we should use elimination.