5.1 Solving Systems of Linear Equations by Graphing

System of Linear Equations

• two linear eqns. considered at the same time• Ex. x + y = 5

x – y = 1

• solutions to systems of eqns. are all ordered pairs that are solns. to BOTH eqns. (both eqns. give a true stmt. when ordered pair is sub. in)

Ex. For the system: x + y = 5x – y = 1

(a) is (3, 2) a soln?

x + y = 53 + 2 = 5 5 = 5 true

x – y = 13 – 2 = 1 1 = 1 true

Since (3, 2) satisfies BOTH eqns, YES, it is a soln to the system

(b) is (-1, 6) a soln?

x + y = 5-1 + 6 = 5 5 = 5 true

x – y = 1-1 – 6 = 1 -7 = 1 false

Since (-1, 6) DOES NOT satisfy BOTH eqns, NO, it is NOT a soln to the system

Solving by Graphing

1. Graph first eqn.2. Graph second eqn. on same set of axes3. Look for a point of intersection

a. The point of intersection is the soln.b. If there is no point of intersection no solutionc. If lines intersect everywhereinfinitely many solns.

4. Check the soln. in BOTH eqns., if necessary

Ex. Solve the system by graphing: y = -2x + 1 x = -1

1) Graph y = -2x + 1y-int: 1, m = -2/1

rise = -2, run = 1

2) Graph x = -1 (vert. line crossing x-axis at -1)

3) Point of intersection is soln. (-1, 3)

4) Check (-1, 3) in both eqns.y = -2x + 1 x = -1

3 = -2(-1) + 1 -1 = -1 3 = 2 + 1 true

3 = 3 true

1 2x

3

1

2

3

-1-2-3

-1

-2

-3

y

Worksheet

Notes:

• Inconsistent system: a system with no soln. (#2 on worksheet)

• Dependent eqns: eqns. that produce the same line (#3 on worksheet)

Lines intersect everywheredependent eqns.Infinite number of

solutions

solution: {(x, y)|eqn.}

Lines (coincide - same line)slope samey-int same

No point of intersectioninconsistent systemNo solution

empty set Ø

Lines (parallel)slope samey-int different

Summary

One point of intersection

solution: {(x, y)}

Lines (distinct lines)slope different