Solving Problems With Two Variables

For instance, \(x = 1\) and \(y = - 4\) will satisfy the first equation, but not the second and so isn’t a solution to the system.

Likewise, \(x = - 1\) and \(y = 1\) will satisfy the second equation but not the first and so can’t be a solution to the system.

\[3x - 7 = y\] Now, substitute this into the second equation.

\[2x 3\left( \right) = 1\] This is an equation in \(x\) that we can solve so let’s do that.

It is quite possible that a mistake could result in a pair of numbers that would satisfy one of the equations but not the other one.

Let’s now move into the next method for solving systems of equations. If fractions are going to show up they will only show up in the final step and they will only show up if the solution contains fractions.\[\begin5\left( \right) 4y & = 1\ 10y \frac 4y & = 1\ 14y & = 1 - \frac = - \frac\ y & = - \left( \right)\left( \right)\ y & = - \frac\end\] Finally, substitute this into the original substitution to find \(x\).\[x = 2\left( \right) \frac = - \frac \frac = \frac\] So, the solution to this system is \(x = \frac\) and \(y = - \frac\).As with single equations we could always go back and check this solution by plugging it into both equations and making sure that it does satisfy both equations.Note as well that we really would need to plug into both equations.The first method is called the method of substitution.In this method we will solve one of the equations for one of the variables and substitute this into the other equation.This means we should try to avoid fractions if at all possible.In this case it looks like it will be really easy to solve the first equation for \(y\) so let’s do that. Due to the nature of the mathematics on this site it is best views in landscape mode.If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.


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