example 1: y = 2x + 1 y = -2x + 5 -...
TRANSCRIPT
Notes: Chapter 6-1 Graphing Systems of Equations
Use a graph to solve a system of equations:
Example 1: y = 2x + 1
y = -2x + 5
Example 2: y = ½x – 1
y = - ½x + 3
Example 3: y = ¾x – 1
y = -¼x + 3
Example 4: y = ⅔x – 1
y = -⅓x + 2
Notes: Ch 6-2 Substitution Method
To solve a system of equations:
Identify the isolated variable in one equation.
Substitute for the same variable in the other equation.
Solve the resulting equation.
Back substitute to solve for the isolated variable.
Use the substitution method to solve for x and y:
Example 1: x = 2y
x + 3y = 10
Example 2: 4x + 2y = 50
y = 3x
Example 3: x = y + 2
3x + y = 10
Example 4: 2x + 4y = 14
y = x – 1
Example 5: x = 2y – 2
3x – 4y = 10
Notes: Chapter 6-2 Substitution Method (Part 2)
To solve a system of equations:
Isolate one variable in either equation.
Substitute for the same variable in the other equation.
Solve the resulting equation.
Back substitute to solve for the isolated variable.
Use the substitution method to solve for x and y:
Example 1: 2x + y = 5
y – x = 4
Example 2: 2x + y = – 4
x + y = – 7
Example 3: 4y – 5x = 9
x – 4y = 11
Example 4: –2x + y = 8
3x + 2y = 9
Example 5: x = 2y – 2
3x – 4y = 10
Notes: Chapter 6-3 Elimination Method
To solve a system of equations:
Arrange both equations in ax + by = c form.
Eliminate a variable by adding like terms.
Solve the resulting equation.
Back substitute to solve for the other variable.
Use the elimination method to solve for x and y:
Example 1: x – y = 3
x + y = 5
Example 2: 5x + 3y = 13
2x – 3y = 1
Example 3: 2x + 3y = 6
– 2x + y = 2
Example 4: x + 2y = 4
x + y = 3
Example 5: 2x – 6y = – 10
2x – 5y = – 9
Notes: Chapter 6-3 Elimination Method with Multiplication
To solve a system of equations:
Arrange both equations in ax + by = c form.
Identify a variable to eliminate.
Multiply one or both equations by numbers so that the
variable to be eliminated has opposite coefficients.
Eliminate that variable by adding like terms.
Solve the resulting equation.
Back substitute to solve for the other variable.
Use the elimination with multiplication method:
Example 1: 5x + 2y = – 30
3x – y = 4
Example 2: 2x + y = 3
– x + 4y = – 6
Example 3: 3x – 5y = – 4
4x + 2y = 12
Example 4: 3x + 2y = 12
2x + 5y = 8
Example 5: 5x + 2y = 7
3x + 7y = 10
Notes: Chapter 6-4 Special Systems of Equations
When a system of equations has at least one solution it is
called a consistent system. If it has exactly one solution it is
called independent if it has more than one solution it is
called dependent.
When a system of equations has no solution it is called an
inconsistent system.
To solve a special system of equations with more than one
solution or no solutions:
Arrange both equations in ax + by = c form.
Eliminate a variable by adding like terms.
The resulting equation will either be always true
(infinitely many solutions) or never true (no solution).
Solve the special system of equations:
Example 1: 7x – y = –2
– 7x + y = 3
Example 2: 5x + 3y = 8
– 5x – 3y = – 8
Example 3: 3x + y = 6
– 9x – 3y = –18
Example 4: y = 2x – 2
– 2x + y = 1
Example 5: y = 2(x + 3)
– 2y = 2x + 6
Notes: Chapter 6-6 Solving Linear Inequalities
A linear inequality has the form
How to Graph a Linear Inequality
Solve the inequality for y.
Graph the boundary line. Use a solid line for ≥ or ≤ . Use a dashed line
for > or < .
Shade the half plane above the line for y > or y ≥ . Shade the half plane for
y < or y ≤ .
Example: Graph the linear inequality.
Example: Graph the linear inequality.
Example: Graph the linear inequality.
Notes: Chapter 6-7 Systems of Inequalities
A system of linear inequalities has the form,
How to Graph a Linear Inequality
Solve each inequality for y.
Graph the boundary line for each inequality. Use a solid line for ≥ or ≤ .
Use a dashed line for > or < .
Shade the half plane above the line for y > or y ≥ . Shade the half plane for
y < or y ≤ .
Example: Graph the system of inequalities.
Example: Graph the system of inequalities.