Section 3.5 Systems of Equations

1
Section 3.5Systems of Equations
2
What is a system of equations?

• Two or more equations in the same variables

3
To solve a system

• Find all ordered pairs (x, y) that make both
  equations true

4
Methods to Solve

1. Graphing method
2. Substitution method
3. Linear combination/Elimination method

5
Solutions

1. One solution, ordered pair (intersection of
   lines)
2. No solution (lines are parallel)
3. Infinitely many solutions (when same exact line)

6
Calculator Directions

• Enter equations in y
• y11st line and y22nd line
• Hit graph to see lines, change window if needed
• Hit 2nd Trace (Calculate)
• Hit or scroll down to Intersect (5)
• Hit enter 3 times to obtain solution
• Write solution as an ordered pair

7
Graphing Method

• Graph each line on the same coordinate plane.
• If lines intersect, there is only one solution
  the intersection point.
• If lines are parallel, there is no solution.
• If lines coincide, there are infinitely many
  solutions.

8
Substitution Method

• Uses substitution of one equation into the other
  to solve for the other variable
• Goal Isolate one variable (if not already
  given)
• Hint Isolate the variable that will allow for
  easy algebra!

9
Linear combination Method

• Add the equations
• Goal To combine the equations to eliminate a
  variable
• Hint Create coefficients that are opposites for
  one of the variables

10
New Vocab

• Consistent Equations- A system of equations with
  at least one solution
• Dependent Equations- A consistent system with
  infinitely many solutions (coinciding lines)
• Inconsistent Equations- A system of equations
  with no solution

11
Use intersect, are parallel, or coincide to make
a true statement

• If two lines have the same slope and different
  y-intercepts, then the lines ___________.
• If one equation can be obtained from another
  equation by multiplying both sides by the same
  nonzero number, then the graphs ___________.

are parallel
coincide
12
Use intersect, are parallel, or coincide to make
a true statement

• If two lines have different slopes and the same
  x-intercept, then the lines __________.
• If two lines have more than one point in common,
  then the lines ___________.
• 5. If the system of equations are dependent,
  then the lines ___________.

intersect
coincide
coincide