Section 3.5Systems of Equations

What is a system of equations?

• Two or more equations in the same variables

To solve a system• Find all ordered pairs (x, y) that make both

equations true

Methods to Solve1. Graphing method

2. Substitution method

3. Linear combination/Elimination method

Solutions1. One solution, ordered pair

(intersection of lines)

2. No solution (lines are parallel)

3. Infinitely many solutions (when same exact line)

Calculator Directions

1. Enter equations in y=• y1=1st line and y2=2nd line

2. Hit graph to see lines, change window if needed

3. Hit 2nd Trace (Calculate)

4. Hit or scroll down to Intersect (#5)

5. Hit enter 3 times to obtain solution

6. Write solution as an ordered pair

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.

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!

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

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

Use intersect, are parallel, or coincide to make a

true statement

1. If two lines have the same slope and different y-intercepts, then the lines ___________.

1. If one equation can be obtained from another equation by multiplying both sides by the same nonzero number, then the graphs ___________.

are parallel

coincide

Use intersect, are parallel, or coincide to make a

true statement3. If two lines have different slopes and the

same x-intercept, then the lines __________.

3. 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