12 terms

6. Systems of Equations and Inequalities

6-1 Solving Systems by Graphing. 6-2 Solving Systems by Substitution. 6-3 Solving Systems by Elimination. 6-4 Solving Special Systems. 6-5 Applying Systems. 6-6 Solving Linear Inequalities. 6-7 Solving Systems of Linear Inequalities.
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system of linear equations
A set of two or more linear equations containing two or more variables.

2x + 3y = −1
x − 3y = 4
solution of a system of linear equations
Any ordered pair that satisfies all the equations in the system.

x + y = −1
−x + y = −3
Solution: (1, −2)
substitution method
A method used to solve systems of equations by solving an equation for one variable and substituting the resulting expression into the other equation(s).

y = 2x
y = x + 5

<b>Step 1</b>
y = 2x
y = x + 5

<b>Step 2</b>
<color blue>y</color> = x + 5
<color blue>2x</color> = x + 5

Step 3
2x <color red>− x</color> = x <color red>− x</color> + 5
x = <color green>5</color>

Step 4
y = 2<color green>x</color>
y = 2<color green>(5)</color>
y = <color purple>10</color>

Step 5
(<color green>5</color>, <color purple>10</color>)

USE WHEN...
• A variable in either equation has a coefficient of 1 or −1.
• Both equations are solved for the same variable.
• Either equation is solved for a variable.

EXAMPLE
x + 2y = 7
x = 10 − 5y

or
x = 2y + 10
x = 3y + 5
elimination method
A method used to solve systems of equations in which one variable is eliminated by adding or subtracting two equations of the system.

x − 2y = −19
5x + 2y = 1

<b>Step 1</b>
<color green> x</color> − <color blue>2y</color> = <color blue>−19</color>
+<color green>5x</color> + <color purple>2y</color> = <color blue>1</color>
∴ <color green>6x</color> + <color purple>0</color> = <color blue>−18</color>

<b>Step 2</b>
6x = −18
6x ÷ <color red>6</color> = −18 ÷ <color red>6</color>
x = <color green>−3</color>

<b> Step 3</b>
<color green>x</color> − 2y = −19
<color green>−3</color> − 2y = −19
<color green>−3</color> <color red>+ 3</color> − 2y = −19 <color red>+ 3</color>
−2y = −16
y = <color purple>8</color>

Step 4
(<color green>−3</color>, <color purple>8</color>)

USE WHEN...
• Both equations have the same variable with the same or opposite coefficients.
• A variable term in one equation is a multiple of the corresponding variable term in the other equation.

EXAMPLE
3x + 2y = 8
5x + 2y = 12

or
6x + 5y = 10
3x + 2y = 15
consistent system
A system of equations or inequalities that has at least one solution.

x + y = 6
x − y = 4
Solution: (5, 1)
inconsistent system
A system of equations or inequalities that has no solution.
independent system
A system of equations that has exactly one solution.

x + y = 7
x − y = 1
Solution: (4, 3)
dependent system
A system of equation that has infinitely many solutions.

x + y = 2
2x + 2 y = 4
linear inequality
An inequality that can be written in one of the following forms: <i>ax < b</i>, <i>ax > b</i>, <i>ax ≤ b</i>, <i>ax ≥ b</i>, or <i>ax ≠ b</i>, where <i>a</i> and <i>b</i> are constants and <i>a</i> ≠ 0.
solution of a linear inequality
An ordered pair or ordered pairs that make the inequality true.

Inequality: 3x + 2y ≥ 6
system of linear inequalities
A set of two or more linear inequalities containing two or more variables.

2x + 3y > −1
x − 3y ≤ 4
solution of a system of inequalities in two variables
Any ordered pair that satisfies all the inequalities in the system.

y ≤ x + 1
y < −x + 4