Solving Systems of Equations

Solving Systems of Equations:

Previously, we have been dealing with just one equation at a time. But now, we will work with more than one variable and more than one equation. These are termed as systems of equations. Whenever answering a system of equations, you require giving the value for each and every variable.

Solving Systems of Linear Equations:

There are six ways that we can utilize to solve a system of the linear equations:


  • Graph both the equations and find out the intersection point.
  • Inaccurate by the hand.
  • Useful whenever using the technology.
  • More suitable for the non-linear systems.
  • Should solve for the equation of y first.


  • Solve one equation for one variable and then replace that to the other equation.
  • Best algebraic method for the non-linear systems.
  • Works well whenever a variable can be solved simply, consists of a coefficient of one.
  • Works better if fractions and roots are not included.


  • Multiply one or more equations by the constant and then add up the two equations altogether to eliminate or remove one variable.
  • Works fine for a linear system if there is no variable with the coefficient of one.
  • Works fine for 2x2 (2 equations with 2 variables) systems of equations; however becomes monotonous and labor intensive for bigger systems.

Gaussian Elimination/Gauss Jordan Elimination:

  • Utilizes the elementary operations to generate equivalent equations.
  • Works for the non-square systems of the linear equations.
  • Built on the concepts of addition-elimination, but rather than obtaining new equations, the old equation is substituted with the equivalent equation.
  • Whenever applied with matrices, probably the fastest method to solve a large system of linear equations by hand. Indeed the instructor's favorite technique.

Cramer's Rule:

  • Utilizes determinants of the matrix to find out the solution.
  • Works just for square systems of linear equations where the determinant of coefficient matrix is not zero.
  • Good for a calculator or computer where there is determinant program.
  • Slow by hand.
  • Slow on calculator devoid of a program as each and every determinant should be entered manually.
  • Can be utilized whenever you need to find out just one of the variables.

Matrix Algebra/Matrix Inverses:

  • Utilizes the inverse of a matrix to find out the solution.
  • Works just for square systems of linear equations where the determinant of coefficient matrix is not zero.
  • Good for a calculator or computer where there is a matrix of inverse function.
  • Slow by hand.
  • Fast to do on calculator.
  • Will return decimal answers, however you can utilize the fraction key to transform it to integers.


The technique of substitution will work with non-linear and also linear equations.

a) Solve one of the equations for one of variables.
b) Replace that expression in for the variable in other equation.
c) Solve the equation for the residual variable.
d) Back-substitute the value for variable to determine the other variable.
e) Check.

The procedure of back-substitution includes taking the value of variable found in step c and replacing it back to the expression obtained in step a (or the original problem) to find out the remaining variable.

It is significant that both the variables be given whenever solving a system of equations. The common mistake students make is to find out one variable and stop there. You need to comprise a value for all variables.

This is a good idea to ensure your answer to the both equations, however is probably adequate to check the equation you did not isolate the variable in first step. That is, when you solved for y in the first equation in step a, employ the second equation to ensure the answer.

Graphical Approach:

The graphical approach works fine with a graphing calculator, however is imprecise by hand (did such points intersect at 1/6 or 1/7?) unless the graph occurs to drop exactly on grid lines.

a) Solve each of equation for y. This might include a plus and minus when there is an y2 term. When you are not graphing with a computer or calculator, you can skip this step.

b) Graph each and every equation.

c) Find out the points of intersection.

d) Check.

It is significant to check your answers to make sure that you read the intersection point properly.

Sometimes the calculator will fail to provide an intersection point by using the intersect command. You might need to use the trace characteristic of the calculator to determine the intersection point. You might use your calculator to ensure the answer.

Try to transform your answer to fractional form when possible.

The graphical approach can keep a lot of time whenever you are working with the non-linear system of equations.

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