System of Linear Equations

 

\pi \cdot \pi
\pi \cdot \pi

\(x\)

\(y\)

\(line~1\)

\(line~2\)

One Solution

\pi \cdot \pi

\(x\)

\(y\)

\(line~1\)

\(line~2\)

One Solution

When solving a system of equations with substitution, if we end in \(x=\#\) or \(y=\#\), there will be one solution. The value of the missing variable can be found by back substitution.

\pi \cdot \pi

\(x\)

\(y\)

\(line~1\)

\(line~2\)

No Solutions

\pi \cdot \pi

\(x\)

\(y\)

\(line~1\)

\(line~2\)

No Solutions

When solving a system of equations with substitution, if we end in no variables and a false statement like \(2=3\), there will be no solutions. The two lines are parallel since they don't intersect.

\pi \cdot \pi

\(x\)

\(y\)

\(line~1\)

\(line~2\)

Infinitely Many Solutions

\pi \cdot \pi

\(x\)

\(y\)

\(line~1\)

\(line~2\)

Infinitely Many Solutions

When solving a system of equations with substitution, if we end in no variables and a true statement like \(4=4\), there will be infinitely many solutions. The two lines are the same since they intersect everywhere.

\pi \cdot \pi

Solving Systems of Linear Equations

  1. Solve any one equation for a variable.

  2. Replace that same variable in the other equation with the step (1) expression.

  3. Solve the equation in step (2) for the remaining variable.

  4. Substitute the variable's value from step (3) in either of the two equations.