# Points, Regions, Distance and Midpoints

**Math 1310
Section 1.1: Points, Regions, Distance and Midpoints**

In this section, we’ll review plotting points , graph
horizontal and vertical lines, some inequalities.

Develop a formula for finding the distance between two points in the coordinate
plane and one for

finding the midpoint of a line segment with given endpoints.

** Graphing Points and Regions**

Here’s the coordinate plane:

As we see the plane consists of two perpendicular lines,
the** x-axis** and the **y-axis**. These two lines separate them

into four regions, or **quadrants.** The pair, (x, y), is called an **
ordered pair .** It corresponds to a single unique

point in the coordinate plane. The first number is called the** x coordinate,**
and the second number is

called the** y coordinate.** The ordered pair (0, 0) is referred to as the **
origin.** The** x coordinate** tells us the

horizontal distance a point is from the origin. The** y coordinate** tells us
the vertical distance a point is

from the origin. You’ll move right or up for positive coordinates and left or
down for negative

coordinates.

Example: Plot the following points.

A. (8,6)

B. (-2,4)

C. (2,5)

D. (-3,-7)

E. (2,-3)

F. (-5,3)

**Graphing Regions in the Coordinate Plane
**

The set of all points in the coordinate plane with y coordinate k is the

**horizontal line y =k .**

The set of all points in the coordinate plane with x coordinate k is the

**vertical line x =k .**

Example: Graph {(x, y)| x > 4 and y ≤ 3 }.

**The Distance Formula**

For any two points and
, the distance between them is given by

Example: Find the distance between the following pair of points.

a) (−3,1) & (1,3)

**Midpoint Formula**

The midpoint of the line segment joining the two points
) and is
given by

Example: Find the midpoint between the following pair of
points.

a) (−3,1) & (1,3)

**Math 1310
Section 1.2: Lines**

In this section, we’ll review slope and different equations of lines. We will also talk about x-intercept and y-intercept,

parallel and perpendicular lines.

**Slope**

Definition: The

**slope**of a line measures the steepness of a line or the rate of change of the line.

To find the slope of a line you need two points. You can find the slope of a line between two points

and by using this formula.

Example 1: Find the slope of the line containing the
following points

a. (4, -3) and (-2, 1)

b. (-3, 1) and (-3, -2)

**Note:
**

-Lines with positive slope rise to the right.

-Line with negative slope fall to the right.

-Lines with slope equal to 0 are horizontal lines.

-Lines with undefined slope are vertical lines

**Finding the Equation of a Line
**

Three usual forms:

1.

**Point-Slope Form**

where
is a point on the line and m is the slope.

2.** Slope- Intercept Form **

y = mx + b

where m is the slope and b is the y-intercept of the line.

3. **Standard Form**

Ax + By + C = 0

where A and B are not both equal to 0.

Example 2: Write the following equation in slope-intercept form and identify the
slope and y-intercept.

2x – 4y =5

Example 3: Write an equation of the line that satisfies the given conditions.

a. m = ½ and the y-intercept is 3.

b. m = -3 and the line passes through (-2, 1).

c. line passes through (-6, 10) and (-2, 2).

**Parallel Lines and Perpendicular Lines
**

Definition: Parallel lines are lines with slopes m

_{1}and m

_{2}such that they are equal, in other words

Definition: Perpendicular lines are lines in which the product of the slopes equal -1.

Also known as the negative reciprocal .

Example 4: Write an equation of the line that passes through the points (-3, 8) and parallel to y= −2x + 4

Example 5: Write an equation of the line that passes
through the points (1, 2) and perpendicular to

y= −2y + 4.

**x-intercept and y-intercept
**

When graphing an equation, it is usually very helpful to find the

**x intercept(s)**and the

**y -intercepts**of the

graph. An x intercept is the first coordinate of the ordered pair of a point where the graph of the equation

crosses the x axis. To find an x intercept, let y = 0 and solve the equation for x.

The

**y-intercept**is the second coordinate of the ordered pair of a point where the graph of

the equation crosses the y axis. To find a y intercept, let x = 0 and solve the equation for

y.

Example 5: Find the x and y intercepts of the graph of the equation 3x - 4y = 8.

Example 6: Find the x and y intercepts of the graph of the equation y = x

^{2}- 9 .

**Math 1310
Section 1.3
Graphing Equations**

One of the things you need to be able to do by the end of this course is to graph several types of equations.

There are many methods to use . In this section, we’ll create a table of values and ordered pairs, and then

plot the points in the coordinate plane. Once we have the points plotted, we can connect the dots to get a

good picture of the equation.

Example 1: Determine which of the points (3, 2), (-1, 3) and (0, 2) are on the graph of the equation 4x - 3y =

6.

Example 2: Determine which of the points (-1, 1), (2, -1)
and (-2, -1) are on the graph of the equation

x^{2}+ 3xy + 2 = 0

When we graphing an equation, it will be helpful to have
more points than just the x and y intercepts of the

graph. We can create a table of values with more choices for x and find the
corresponding y values.

Example 3: Sketch the graph of the equation by plotting points: y = -3x + 2 .

Example 4: Sketch the graph of the equation by plotting points: y = x + 3

**Math 1310
Section 2.1: Linear Equations **

Definition: To solve an equation in the variable x using the

**algebraic method**is to use the rules of algebra

to isolate the unknown x on one side of the equation.

Definition: To solve an equation in the variable x using the

**graphical method**is to move all terms to one

side of the equation and set those terms equal to y. Sketch the graph to find the values of x where

y = 0.

Example 1: Solve the following equation

**algebraically.**

5y + 6 = −18 − y

Example 2: Solve following equation **algebraically.
**

7 + 2(3 – 8x) = 4 – 6(1 + 5x)

Example 3: Solve following equation

**algebraically**

Example 4: Solve following equation **algebraically.**

Example 5: Find the x-intercept and y-intercept of the
following equation. Express the answers in

coordinate point form.

**Math 1310
Section 6.1: Solving 2x2 Linear Systems**

To solve a system of two linear equations

means to find values for x and y that satisfy both
equations.

The system will have exactly one solution , no solution, or infinitely many
solutions.

1. Exactly one solution, will look like:

2. No solution, will look like:

3. Infinitely many solutions, will look like:

Example 1: Solve the following systems of linear equations
by the substitution method.

2x – y = 5

5x + 2y = 8

Example 2 : Solve the following systems of linear equations by the substitution
method

x – 2y = 3

2x – 4y = 7

Example 3: Solve the following systems by the Elimination
Method .

2x + 3y = -16

5x – 10y = 30

Example 4: Solve the following systems by the Elimination Method.

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