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Find the values of [tex]$x$[/tex] and [tex]$y$[/tex] from the following equations:

[tex]\[
\begin{array}{l}
3x - 2y = 4 \\
x + y = 2
\end{array}
\][/tex]

A. [tex]$x = 2$[/tex] and [tex]$y = 4$[/tex]
B. [tex][tex]$x = 4$[/tex][/tex] and [tex]$y = 6$[/tex]
C. [tex]$x = \frac{3}{2}$[/tex] and [tex][tex]$y = \frac{2}{3}$[/tex][/tex]
D. [tex]$x = \frac{8}{5}$[/tex] and [tex]$y = \frac{2}{5}$[/tex]
E. [tex][tex]$x = \frac{4}{3}$[/tex][/tex] and [tex]$y = \frac{3}{5}$[/tex]


Sagot :

To solve the system of equations:
[tex]\[ \begin{array}{l} 3x - 2y = 4 \\ x + y = 2 \end{array} \][/tex]

we will use the substitution or elimination method. Here, I'll demonstrate using the substitution method.

1. Solve the second equation for [tex]\( x \)[/tex]:
[tex]\[ x + y = 2 \implies x = 2 - y \][/tex]

2. Substitute [tex]\( x = 2 - y \)[/tex] into the first equation:
[tex]\[ 3(2 - y) - 2y = 4 \][/tex]

3. Simplify and solve for [tex]\( y \)[/tex]:
[tex]\[ 6 - 3y - 2y = 4 \\ 6 - 5y = 4 \\ -5y = 4 - 6 \\ -5y = -2 \\ y = \frac{2}{5} \][/tex]

4. Substitute [tex]\( y = \frac{2}{5} \)[/tex] back into the equation [tex]\( x = 2 - y \)[/tex]:
[tex]\[ x = 2 - \frac{2}{5} \\ x = \frac{10}{5} - \frac{2}{5} \\ x = \frac{8}{5} \][/tex]

Thus, the values are:
[tex]\[ x = \frac{8}{5}, \quad y = \frac{2}{5} \][/tex]

So, the correct choice is [tex]\( \boxed{D} \)[/tex].