Solve Systems By Graphing

Definitions:

·  Dependent System: system does not have a unique solution

·  Inconsistent System: system that has no solution

·  Independent System: system having a unique solution

·  System of Linear Equations: 2 or more linear equations

·  System of Linear Inequalities: 2 or more linear inequalities

·  Solution of the System of Linear Equations: any ordered pair in a system that makes all equations true.

·  Solution of the System of Linear Inequalities: any ordered pair in a system that makes all inequalities true.

Key Concepts:

·  One Solution: Different Slopes (independent)

·  No Solution: Lines are parallel, same slopes but different y-intercepts (inconsistent)

·  Infinitely Many Solutions: Lines are the same, same slope and same y-intercepts (dependent)

Examples: Determine # of Solutions without Graphing

Without graphing, determine whether each system has 1 solution, no solution or infinitely many solutions. Then classify each system as independent, dependent or inconsistent.

1.  y = -2x + 1 b=(0,1) m = 2. y = x b=(0,0) m =

y = -2x –1 b= (0,-1) m = 3x –5y = 0 → y = x b =(0,0) m=

Same slope, different y-intercepts same slope, same y-intercepts

Therefore parallel lines -> no solution Therefore, same line-> infinitely many

Inconsistent Dependent

Practice: Determine # of Solutions without Graphing

Without graphing, determine whether each system has 1 solution, no solution or infinitely many solutions. Then classify each system as independent, dependent or inconsistent.

3.  y = x + 9 4. y = 3x + 2

5y = x + 45 y = 3x - 2

5.  7x – y = 6 6. x – 3y = 2

-7x + y = -6 4x – 12y = 8

Solving System of Equations by Graphing

In order to solve a system of equations by graphing

·  Slope-Intercept Form (y = mx + b):

o  Graph each equation using the y-intercept(b) and the slope (m).

·  Standard Form (Ax + By = C)

o  Find the slope (-A/B)

o  Find the x-intercepts & y-intercepts

o  Graph using the slope, x-intercepts & y-intercepts

·  In order to ensure that both lines are long and will connect, make sure to plot the slope of each line several times in both directions.

·  The point where 2 lines intersect is the solution the system.

·  Check to see if the solution makes both equations true.

Examples: Solve System of Equations by Graphing

Solve by graphing. Check your solution.

1.  y = 2x – 3
y = x – 1
Slope Intercept Form:
Step 1: Find slope & y-intercepts
b1 = (0, -3) m1 = (2↑ 1→)
b2 = (0, -1) m2 = (1↑ 1→)
Step 2: Determine # of solutions
Since slopes and y-intercepts are different, there is 1 solution. It is independent. / Step 3: Graph equations & look for point of intersection
Step 4: Check solution (2, 1)
y = 2x – 3 y = x – 1
1 = 2(2) –3 1 = 2 – 1
1= 4 – 3 1 = 1
1 = 1
2.  –x + y = 5
4x + y = 0
Standard Form:
Step 1: Find slope, x-intercepts & y-intercepts
b1 = (0, 5) m1 = (1↑ 1→)
b2 = (0, 0) m2 = (4↓1→)
Step 2: Determine # of solutions
Since slopes and y-intercepts are different, there is 1 solution. It is independent. / Step 3: Graph equations & look for point of intersection
Step 4:
Check solution (-1, 4)
-x + y = 5 4x + y = 0
-(-1)+ 4 = 5 4(-1) + 4 = 0
1 + 4 = 5 -4 + 4 = 0

Practice: Solve System of Equations by Graphing

Solve by graphing. Check your solution.

1.  y = 2x + 1 2. y = x + 2

y = 3x – 1 y = -2x + 2

Examples: Word Problem

Suppose you have $20 in your bank account. You start saving $5 each week. Your friend has $5 in his account and is saving $10 each week. Assume that neither you nor your friend makes any withdrawals.

a.  After how many weeks will you and your friend have the same amount of money in your accounts?

b.  How much money will each of you have?

Solving System of Inequalities by Graphing

In order to solve a system of inequalities by graphing

·  Repeat the same steps for graphing equations

·  Use the inequality symbol to draw either dotted or solid lines and shade the appropriate area.

·  The area that is shaded by both inequalities is the solution to the system of inequalities.

·  Check to see if the solution makes both inequalities true.

Example: Solve System of Inequalities by Graphing

Solve by graphing. Check your solution.

1.  y > 2x – 5
2x + 1y < 12
Step 1: Find relevant information to graph
Ineq 1: Slope Intercept Form:
Find slope & y-intercepts
b1 = (0, -5) m1 = (2↑ 1→)
Ineq 2: Standard Form:
Find slope, x-intercepts & y-intercepts
b2 = (0,3) m2 = - (2↓ 1→), x-int = (6,0) / Step 2: Graph inequalities & look for overlapping region
Step 3: Find a solution & check it.

Practice: Solve System of Inequalities by Graphing

Solve by graphing.

1. y < 2x + 4 2. y > x

2x - y ≤ 4 y ≤ -x + 4

3. y ≥ -x + 5 4. y ≥ -x + 1

y ≤ 3x - 4 y ≤ -x + 7

5.  y ≥ -|x| + 5 6. y ≥ -|x + 1|

y ≤ 4 y ≤ -x + 7

3 Rev B