Unit 5 – Systems of Linear Equations

Unit 5.1 – Systems of Linear Equations in Two Variables

A system of linear equations in two variables is simply two or more linear equations that are solved simultaneously to find the solution set. Because we are solving a system (multiple equations) we are looking for the solution set that contains all ordered pairs that solve all of the equations in the system at the same time.

Solve a system by Graphing

Graph each equation and find the point where the graphs intersect.

Example:

To be sure that (3, 2) is a solution to both equations, substitute it into each equation and solve.

Is an ordered pair a solution of a linear system?

Examples:

Decide whether the given ordered pair is a solution to the system

; (4, 2)
/ / Since (4, 2) makes both equations true, it is a solution to the system.
; (-1, 7)
/ / Since (-1, 7) does not make both equations true, it is not a solution to the system.

Graphs of linear systems in two variables

  1. The two graphs intersect at one point. The system is consistent and the equations are independent.
  1. The graphs are parallel lines. The system is inconsistent, no common solution.
  1. The graphs are the same line. The equations are dependent, since any solution of one equation is also a solution of the other equation.

Solving a system by Substitution

  1. Solve one of the equations for either variable.
  2. Substitute for that variable in the other equation.
  3. Solve the new equation from Step 2.
  4. Find the other variable. Substitute the result from Step 3 into the equation from Step 1 and solve.
  5. Check the solution

Example:

Step 1: Solve one equation for x or y. We’ll solve the second one for y. /
Step 2: Substitute for that variable in the other equation. /
Step 3: Solve the new equation from Step 2. /
Step 4: Find the other variable. Substitute the result from Step 3 into the equation from Step 1 and solve. /
Step 5: Check your answer. / /

Solving a system by Elimination

  1. Write both equations in standard form.
  2. Make the coefficients of one pair of variable terms opposites.
  3. Add the new equations to eliminate a variable.
  4. Solve the equation from Step 3 for the remaining variable.
  5. Find the other value. Substitute the result of Step 4 into either of the original equations and solve.
  6. Check the solution.

Example:

Step 1: Write both in standard form. / They already are.
Step 2: Let’s eliminate x. Multiply the first equation by 2 and the second by -5. /
Step 3: Now add the equations together. /
Step 4: Solve for y. /
Step 5: Find the other value. Substitute the result of Step 4 into either of the original equations and solve. /
Step 6: Check your answer. / /

Special cases of Linear Systems

If both variables are eliminated when a system of linear equations is solved, then

  1. there is no solution if the resulting statement is false.
  2. there are infinitely many solutions if the resulting statement is true.

Unit 5 - Systems of Linear Equations - Student NotesPage 1 of 4