Solving Systems of Linear Equations Using Substitution
Systems of Linear equations:
A system of linear equations is just a set of two or more linear equations.
In two variables , the graph of a system of two equations is a pair of lines in the plane.
There are three possibilities:
- The lines intersect at zero points. (The lines are parallel.)
- The lines intersect at exactly one point. (Most cases.)
- The lines intersect at infinitely many points. (The two equations represent the same line.)
How to Solve a System Using The Substitution Method
- Step : First, solve one linear equation for in terms of .
- Step : Then substitute that expression for in the other linear equation. You'll get an equation in .
- Step : Solve this, and you have the -coordinate of the intersection.
- Step : Then plug in to either equation to find the corresponding -coordinate.
Note : If it's easier, you can start by solving an equation for in terms of , also – same difference!
Solve the system
Solve the second equation for .
Substitute for in the first equation and solve for .
Substitute for in and solve for .
The solution is .
Note : If the lines are parallel, your -terms will cancel in step , and you will get an impossible equation, something like .
Note : If the two equations represent the same line, everything will cancel in step , and you will get a redundant equation, .