How do you solve a system of linear equations using substitution or elimination?

Given a system of equations, there are a few methods to calculate solutions of that system. Two ways to do so are by using elimination or substitution.

To solve a set of equations by elimination you will take two equations and either add or subtract them to eliminate one of the variables. Here is a quick example...

[tex]\left \{ {{3x+y=5} \atop {2x-y=0}} \right.[/tex]

If we were to add these equations together, we could eliminate the variable [tex]y\\[/tex] to get an equation to solve for [tex]x[/tex].

After adding these equations we get: [tex]5x=5[/tex]

We then can solve the equation for [tex]x\\[/tex], to find the value of [tex]x[/tex], and use that value to plug back (a.k.a substitute) into the other equations to solve for [tex]y[/tex]

To solve a set of equations by substitution you will take a system of equations, pick one of the equations and solve one of them for one variable. Here is a quick example...

[tex]\left \{ {{3x+2y=16} \atop {7x+y=19}} \right.[/tex]

If we take the second equation and solve for the variable, [tex]y[/tex], we will get an equation in terms of [tex]x[/tex]. We can then take that equation and plug it into the top, substituting [tex]y[/tex], for the equation in terms of [tex]x[/tex]. Like so....

Solving bottom equation for [tex]y[/tex], we get: [tex]y=19-7x[/tex], now substitute this equation for [tex]y[/tex] into the top equation.

We get: [tex]3x+2(19-7x)=16[/tex], you now have an equation only in terms of [tex]x[/tex], so you can solve for [tex]x[/tex]. I won't complete the whole problem but hopefully you get the idea :)