1. Systems of Linear Equations and Inequalities

2. The length is one more than twice the width.
P = 26 inches
The length is one more than twice the width.
Let x = width, y = length
2x + 2y = 26
y = 2x + 1
(4, 9)

3. Solving a System of Linear Equations, Method 1
Substitution
Step 1: Solve one equation for x (or for y).
Step 2: Substitute the value you found in Step 1 into the other equation.
The result is an equation with one variable.
Step 3: Solve this new equation for y (or for x).
Step 4: Use the value you found in Step 3 for y (or for x) and substitute it
into one of the original equations. Solve for x (or for y).
Given the system:
x + 3y = 12
2x + y = 9
Use substitution to write the solution as an ordered pair (x, y).

4. Solving a System of Linear Equations, Method 2
Elimination
Step 1: Choose one variable to eliminate.
Step 2: Choose a multiplier for one or both equations, so that the
equations have opposite terms. Multiply by the multiplier(s).
Step 3: Add the two equations. One variable will be eliminated.
Step 4: Solve for the other variable.
Use elimination to write the solution as an ordered pair (x, y).
Given the system:
x + 3y = 12
2x + y = 9
Step 5: Replace the solved variable with its value.
Solve for the remaining variable.

5. Solving a System of Linear Equations, Method 3
Graphing
Step 2: Graph both equations on the same coordinate grid.
Step 3: Locate the point(s) where the two lines intersect.
Use graphing to find the solution.
Given the system:
x + 3y = 12
2x + y = 9
Step 1: Write both equations in slope-intercept form.

6. Classifying Systems of Equations
consistent — has at least one solution
dependent — has infinite solutions
independent — has exactly one solution
inconsistent — has no solution
Classify the systems as
dependent consistent, independent consistent, or inconsistent
3x + y = 9
6x + 2y = 18
5 – 2y = x
x + 3y = 6
y = 3x + 7
y – 1 = 3x

7. Classifying Systems of Equations
consistent — has at least one solution
dependent — has infinite solutions
independent — has exactly one solution
inconsistent — has no solution
Classify the systems as
dependent consistent, independent consistent, or inconsistent
3x + y = 9
6x + 2y = 18
5 – 2y = x
x + 3y = 6
y = 3x + 7
y – 1 = 3x
dependent consistent
independent consistent
inconsistent

8. Classifying Graphs of Systems of Equations
dependent consistent
independent consistent
inconsistent
infinite
one
none
same
different
same or different
Number of Solutions
Slopes
y-intercepts
Classification
Type of Graph

9. Solving a System of Linear Inequalities
Step 1: Choose one variable to eliminate.
Step 2: Choose a multiplier for one or both equations, so that the
Step 3: Add the two equations. One variable will be eliminated.
Step 4: Solve for the other variable.
Graph to find the solution region.
Given the system:
x + 3y ≥ 12
2x + y < 9
Step 5: Replace the solved variable with its value.
equations have opposite terms. Multiply by the multipliers.
Solve for the remaining variable.

10. Solving a System of Linear Inequalities
Graph to find the solution region.
Given the system:
x + 3y ≥ 12
2x + y < 9
Step 1: Choose one variable to eliminate.
Step 2: Choose a multiplier for one or both equations, so that the
Step 3: Add the two equations. One variable will be eliminated.
Step 4: Solve for the other variable.
Step 5: Replace the solved variable with its value.
Solve for the remaining variable.
SOLUTION REGION
equations have opposite terms. Multiply by the multipliers.

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