5.1 - Solving Systems of Equations

Up until this point, we have been dealing with only one equation at a time. Now, we will work with more than variable and more than one equation. These are called systems of equations. When answering a system of equations, you need to give the value for each variable.

Solving Systems of Linear Equations

When we're through covering the two chapters on solving systems of equations, there will be six ways that we can use to solve a system of linear equations

Graphically
Graph both equations and find the intersection point.
Inaccurate by hand.
Useful when using technology.
More appropriate for non-linear systems.
Must solve for the equation for y first.
Substitution
Solve one equation for one variable and then substitute that into the other equation.
Best algebraic technique for non-linear systems.
Works well when a variable can be solved for easily, has a coefficient of one.
Works better when fractions and roots aren't involved.
Addition / Elimination
Multiply one or more equations by a constant and then add the two equations together to eliminate one variable.
Works well for a linear system when there is no variable with a coefficient of one.
Works well for 2x2 (2 equations with 2 variables) systems of equations, but becomes tedious and labor intensive for larger systems.
Gaussian Elimination / Gauss Jordan Elimination
Uses elementary operations to produce equivalent equations.
Works for non-square systems of linear equations.
Built upon the concepts of addition elimination, but instead of obtaining new equations, the old equation is replaced with an equivalent equation.
When applied with matrices from chapter 6, probably the fastest way to solve a large system of linear equations by hand. Certainly the instructor's favorite method.
Cramer's Rule
Uses determinants of a matrix to find the solution.
Works only for square systems of linear equations where the determinant of the coefficient matrix isn't zero.
Good for a computer or calculator where there is a determinant program.
Slow by hand.
Slow on the calculator without a program since every determinant must be entered manually.
Can be used when you need to find just one of the variables.
Matrix Algebra / Matrix Inverses
Uses the inverse of a matrix to find the solution.
Works only for square systems of linear equations where the determinant of the coefficient matrix isn't zero.
Good for a computer or calculator where there is a matrix inverse function.
Slow by hand.
Quick to do on the calculator.
Will return decimal answers, but you can use the fraction key to convert it to integers.

Substitution

The method of substitution will work with non-linear as well as linear equations.

  1. Solve one of the equations for one of the variables.
  2. Substitute that expression in for the variable in the other equation.
  3. Solve the equation for the remaining variable
  4. Back-substitute the value for the variable to find the other variable.
  5. Check

The process of back-substitution involves taking the value of the variable found in step 3 and substituting it back into the expression obtained in step 1 (or the original problem) to find the remaining variable.

It is important that both variables be given when solving a system of equations. A common mistake students make is to find one variable and stop there. You need to include a value for all the variables.

It is a good idea to check your answer into the both equations, but is probably sufficient to check in the equation you didn't isolate a variable in the first step. That is, if you solved for y in the first equation in step 1, use the second equation to check the answer.

Graphical Approach

The graphical approach works well with a graphing calculator, but is inaccurate by hand (did those points intersect at 1/6 or 1/7?) unless the graph happens to fall exactly on the grid lines.

  1. Solve each equation for y. This may involve a plus and minus if there is a y2 term. If you're not graphing with a calculator or computer, you can skip this step.
  2. Graph each equation.
  3. Find the points of intersection.
  4. Check!

It is important to check your answers to make sure that you have read the intersection point correctly.

Sometimes the calculator will fail to give an intersection point using the intersect command. You may need to use the trace feature of the calculator to find the intersection point. You may use your calculator to check the answer.

Try to convert your answer to fractional form if possible.

The graphical approach can save a lot of time when you're working with a non-linear system of equations.