Solving Systems of Linear Equations

Methods

  • Graphing
  • Substitution
  • Elimination

Solving Systems of Linear Equations

Note: The above video is all the ways not just graphing.

FAQ

When do I solve by graphing?

Use when:

  • Both equations are solved for y.
  • You want to estimate a solution.
When do I solve using substitution?

Use when:

  • A variable in either equation has a coefficient of 1 or -1.
  • Both equations are solved for the same variable.
  • Either equation is solved for a variable.
When do I solve using elimination?

Use when:

  • Both equations have the same variable with the same or opposite coefficients.
  • A variable term in one equation is a multiple of the corresponding variable term in the other equation.

Examples

Graphing

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Substitution

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Elimination

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Methods Pros and Cons

Substitution

This method involves plugging an expression from one equation in for the variable in another. To use this method, at least one variable in one of the equations must be isolated. This is why substitution is most useful when the problem already contains an isolated variable or if there is at least a variable that has a coefficient of one. If you can solve basic algebra equations very quickly, substitution is a good choice. However, it poses problems for those who tend to make arithmetic mistakes.

Elimination

To use elimination, you must line up both equations vertically with the variables on one side and constants on the other. The bottom equation is then subtracted from the top one to cancel out a variable. This makes elimination efficient when the constants of both equations are already isolated. Additionally, if the coefficients of the Xs or Ys in both equations are the same, elimination will get a solution quickly with minimal steps. On the other hand, sometimes one or both whole equations have to be multiplied by a number to make the variable cancel. This can make the work take longer, and elimination is not the best choice in this scenario.

Graphing by Hand

If the equations do not involve fractions or decimals, and you have a good visual understanding of linear equations, graphing on the coordinate plane is a good option. This technique involves visually finding the point on the graph where the two lines cross to get the solutions for X and Y. Because it helps you to graph quickly, having both equations in Y= form makes this method useful. In contrast, if neither equation has Y isolated, you are better off using substitution or elimination.

Graphing on a Calculator

Using a graphing calculator to enter both equations and find the point of intersection comes in handy when they involve decimals or fractions. It is also a good choice when the teacher allows such calculators on tests or quizzes. However, as in graphing by hand, this technique works best when the Ys in both equations are already isolated.

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