12 terms

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.

system of linear equations

A set of two or more linear equations containing two or more variables.

2x + 3y = −1

x − 3y = 4

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)

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

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

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)

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)

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

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

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

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

y ≤ x + 1

y < −x + 4