# 7.9: Summary of Key Concepts

- Page ID
- 60040

## Summary of Key Concepts

**Graph of a Function **

The geometric representation (picture) of the solutions to an equation is called the **graph** of the equation.

**Axis **

An **axis** is the most basic structure of a graph. In mathematics, the number line is used as an axis.

**Number of Variables and the Number of Axes**

An equation in one variable requires one axis | One-dimension |

An equation in two variables requires two axes. | Two-dimensions |

An equation in three variables requires three axes. | Three-dimensions. |

An equation in \(n\) variables requires \(n\) axes | \(n\)-dimensions. |

**Coordinate System **

A system of axes that is constructed for graphing an equation is called a **coordinate system**.

**Graphing an Equation**

The phrase **graphing an equation** is interpreted as meaning geometrically locating the solutions to that equation.

**Uses of a Graph **

A graph may reveal information that may not be evident from the equation.

**Rectangular Coordinate System \(xy\)-Plane **

A **rectangular coordinate system** is constructed by placing two number lines at 90∘ angles. These lines form a plane that is referred to as the \(xy\)-plane.

**Ordered Pairs and Points**

For each ordered pair \((a,b)\), there exists a unique point in the plane, and for each point in the plane we can associate a unique ordered pair \((a,b)\) of real numbers.

**Graphs of Linear Equations **

When graphed, a linear equation produces a straight line.

**General Form of a Linear Equation in Two Variables **

The **general form** of a linear equation in two variables is \(ax+by=c\), where \(a\) and \(b\) are not both \(0\).

**Graphs, Ordered Pairs, Solutions, and Lines **

The graphing of all ordered pairs that solve a linear equation in two variables produces a straight line.

The graph of a linear equation in two variables is a straight line.

If an ordered pair is a solution to a linear equation in two variables, then it lies on the graph of the equation.

Any point (ordered pair) that lies on the graph of a linear equation in two variables is a solution to that equation.

**Intercept **

An **intercept** is a point where a line intercepts a coordinate axis.

**Intercept Method **

The **intercept method** is a method of graphing a linear equation in two variables by finding the intercepts, that is, by finding the points where the line crosses the \(x\)-axis and the \(y\)-axis.

**Slanted, Vertical, and Horizontal Lines **

An equation in which both variables appear will graph as a **slanted** line.

A linear equation in which only one variable appears will graph as either a **vertical** or **horizontal** line.

\(x=a\) graphs as a vertical line passing through \(a\) on the \(x\)-axis.

\(y=b\) graphs as a horizontal line passing through \(b\) on the \(y\)-axis.

**Slope of a Line **

The slope of a line is a measure of the line’s steepness. If \((x_1,y_1)\) and \((x_2,y_2)\) are any two points on a line, the slope of the line passing through these points can be found using the slope formula.

\(m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{\text{ vertical change }}{\text{ horizontal change }}\)

**Slope and Rise and Decline **

Moving left to right, lines with positive slope rise, and lines with negative slope decline.

**Graphing an Equation Given in Slope-Intercept Form **

An equation written in slope-intercept form can be graphed by

- Plotting the \(y\)-intercept \((0,b)\).
- Determining another point using the slope, \(m\).
- Drawing a line through these two points.

**Forms of Equations of Lines**

General Form: \(ax + by + c\)

Slope-Intercept Form: \(y = mx + b\)

To use this form, the slope and \(y\)-intercept are needed

Point-Slope Form: \(y - y_1 = m(x - x_1)\)

To use this form, the slope and one points, or two points, are needed.

**Half-Planes and Boundary Lines **

A straight line drawn through the plane divides the plane into two **half-planes**. The straight line is called a **boundary line**.

**Solution to an Inequality in Two Variables **

A solution to an inequality in two variables is a pair of values that produce a true statement when substituted into the inequality.

**Location of Solutions to Inequalities in Two Variables **

All solutions to a linear inequality in two variables are located in one, and only one, half-plane.