Answer :
Let x and y be the first and second number, respectively. Then, the statement "adding 4 times the first number and 7 times the second number gives a total of -9" can be written as
[tex]4x+7y=-9[/tex]and the statement "adding 3 times the first number and 9 times the second number gives -3" can be written as
[tex]3x+9y=-3[/tex]Therefore, the system of equations is:
[tex]\begin{gathered} 4x+7y=-9 \\ 3x+9y=-3 \end{gathered}[/tex]Solving by elimination method.
By multiplying by -3 the first equation and by 4 the second one, we obtain an equivalent system of equations:
[tex]\begin{gathered} -12x-21y=27 \\ 12x+36y=-12 \end{gathered}[/tex]Then, by adding both equations, we get
[tex]\begin{gathered} 36y-21y=27-12 \\ \text{which gives} \\ 15y=15 \end{gathered}[/tex]So, y is given by
[tex]\begin{gathered} y=\frac{15}{15} \\ y=1 \end{gathered}[/tex]Finally, we can find x by substituting this value into the second equation of our system. It yields,
[tex]3x+9(1)=-3[/tex]which gives
[tex]\begin{gathered} 3x+9=-3 \\ 3x=-12 \\ \text{then} \\ x=\frac{-12}{3} \\ x=-4 \end{gathered}[/tex]Therefore, the two numbers are: 1 and -4.