Given the equation system
[tex]\begin{gathered} 2x+3y=9 \\ x-2y=1 \end{gathered}[/tex]First step is to write of the equations in terms of one of the variables, for example I'll write the second equation in terms of x:
[tex]\begin{gathered} x-2y=1 \\ x=1+2y \end{gathered}[/tex]Second step is to replace the expression obtained for x in the first equation:
[tex]2(1+2y)+3y=9[/tex]Solve the parentheses using the distributive property of multiplications:
[tex]\begin{gathered} 2\cdot1+2y\cdot2+3y=9 \\ 2+4y+3y=9 \\ 2+7y=9 \end{gathered}[/tex]Pass the 2 tothe other side of the equation by subtracting it from both sides of the = sign
[tex]\begin{gathered} 2-2+7y=9-2 \\ 7y=7 \end{gathered}[/tex]And divide both sides by 7
[tex]\begin{gathered} \frac{7y}{7}=\frac{7}{7} \\ y=1 \end{gathered}[/tex]Replace it in the first expression obtained to determine the value of x:
[tex]\begin{gathered} x=1-2y \\ x=1-2\cdot1 \\ x=1-2 \\ x=-1 \end{gathered}[/tex]This sistem of equations has one solution at x=-1 and y=1
