Question
Mathematics
$\begin{array}{l}4x-2y=12\\ -10x+5y=-30\end{array}${:[4x-2y=12],[-10 x+5y=-30]:}
Solve problem with AI
We can simplify the first equation by factoring out a 2:
$\begin{array}{rl}4x-2y& =12\\ \\ 2\left(2x-y\right)& =12\\ \\ 2x-y& =6\\ \end{array}${:[4x-2y=12],[],[2(2x-y)=12],[],[2x-y=6],[]:}
Now we have the system:
$\begin{array}{rl}2x-y& =6\\ \\ -10x+5y& =-30\end{array}${:[2x-y=6],[],[-10 x+5y=-30]:}
We can simplify the second equation by factoring out a 5:
$\begin{array}{rl}-10x+5y& =-30\\ \\ 5\left(-2x+y\right)& =-30\\ \\ -2x+y& =-6\\ \end{array}${:[-10 x+5y=-30],[],[5(-2x+y)=-30],[],[-2x+y=-6],[]:}
Now we have the system:
$\begin{array}{rl}2x-y& =6\\ \\ -2x+y& =-6\end{array}${:[2x-y=6],[],[-2x+y=-6]:}
We can solve the second equation for $y$y:
$\begin{array}{rl}-2x+y& =-6\\ \\ y& =2x-6\\ \end{array}${:[-2x+y=-6],[],[y=2x-6],[]:}
Now we can substitute this expression for $y$y into the first equation and solve for $x$x:
$\begin{array}{rl}2x-y& =6\\ \\ 2x-\left(2x-6\right)& =6\\ \\ 2x-2x+6& =6\\ \\ 6& =6\\ \end{array}${:[2x-y=6],[],[2x-(2x-6)=6],[],[2x-2x+6=6],[],[6=6],[]:}
This equation is true for all values of $x$x because we have an identity. Therefore, there are infinitely many solutions to the system. We can write the solution set as $\left(x,y\right)=\left(x,2x-6\right)$(x,y)=(x,2x-6), where $x$x can be any real number.
You might be interested in...
Explore more...