Starring:Je Olive Kathleen Ballener

Jasmine Montes

Yani Mae P itaKiesheen May Martonia

Marisol Aguilar Honey Grace Tinaco

Dianne Joy Cosares

5.1 “The Cartesian Coordinate Plane “

These two number lines make up what

we call the coordinate plane as shown

here. The number lines intersect at the

point called the origin denoted by the

letter O. The horizontal number line called

the x-axis and the vertical number

line is called the y-axis. ArrowheadsAt each end of both axes indicate the infinity of the set of real numbers. Notice that the axes divide theplane into four regions or quadrants labeled with Roman numerals. I through IV in

counterclockwise direction. The first number x is called x coordinate or abscissa and the second number y is called the y Coordinate or ordinate.

The plane described is often called the rectangular coordinate system or CartesianCoordinate System. The word Cartesian is used in Rene Descartes,

4

3

2

1

-1

-2

-3

-4

-1 -2 -3 -4 1 2 3 4

y

x

I 11

111 1V

the 17th-century French philosopher and mathematician who first devised the coordinate system.

5.25.2 ““Points in the Cartesian Points in the Cartesian Coordinate Plane”Coordinate Plane”

There exist a one-to-one correspondence between points in the plane and the ordered pairs of real numbers.

5.2.1 “The coordinates of a point”

The distance of a point from the x- and y-axes is measured in units from the point along the line perpendicular to the respective axis.

5.2.2 5.2.2 “Plotting of Points”“Plotting of Points”

If points in a coordinate plane canbe named. Points can also be plotted in the plane given their coordinates.

To locate the P represented by the ordered pair (3,4). Point the tip of your pencil at the origin (0,0) move it 3 units to the right, then 4 units upward yourpoint is at P is 3 andthe y-coordinate x-coordinate is 4. The processof locating a point in the coordinate plane is called plotting the point.

5.2.3 “Points in a Quadrants”

Given coordinates of a point, the quadrant where it is located can be determined. In a quadrant I, both abscissa and ordinate are positive (x, y). In quadrant II,the abscissa is negative while the ordinateis positive, (x, y). In quadrant III, both abscissa and ordinate are negative, (x, y).

Points found on any of the axes are not considered to be in any

quadrants.

y

LINEAR EQUATIONS IN TWO LINEAR EQUATIONS IN TWO VARIABLES : VARIABLES : Ax + By = CAx + By = C

The graph of a line can be drawn using ordered pairs of numbers in

the form (x, y). The abscissa is the

x-coordinate and the ordinate is

the y-coordinate.                                                                                      

     

The ordered pairs (-1,4),(0,2),(1,0),(2,-2)

are some of its solutions, that is substituting each pair in the sentence

will give a true statement. If the domain is the set of real numbers, the paragraph is a line. This line

is the graph of the equation and the sentence is the equation of the line. The mathematical sentence 2x + y = 2 is an example of a linear equation.

5.35.3 “The graph of Ax + by = “The graph of Ax + by = c based on the table of c based on the table of

values”values”

Values and representing the domain (x) and the range (y), the graph an equation can drawn.

5.3.2 “Intercepts, Slope, Domain, and Range”

A.Study the graph of the equation 3x – 2y = 6

Notice that the line crosses the x- and y- axes at (2, 0) and (0, -3).The x-intercept is the abscissa of the point (2, 0) where the graph crosses the x- axis. They y- intercept is

the ordinate of the point (0, -3) where the graph crosses the y- axis.

3x + 2y = 6

x(2, 0)

(0, -3)

(-1,4) 4

2 (0,2)

(1,0)

(2,-2)

2x + y = 2

y

5.3.3 “Properties of the Graph of a Linear Equation Ax + By = C”

The graph of every equation of the form Ax + By = C, where a and b are not both zero, is a line. In studying of this equation, there are properties of the graph that have to be considered.

5.3.3a “The Intercepts”

The set of ordered pairs which satisfies the equation are (-3, 4), (3, 0), (6, -2).The point whose coordinates consist the number pair (0, 2) intersects the y-axis, thus the

ordinate 2 point whose coordinates consist of the number pair (3, 0) intersects the x-axis, thus the abscissa 3 of this point is called the x-intercepts.

Remember:The x- intercept is the abscissa of the point (a, 0)Where a graph intersects the axis.

The y-intercept is the ordinate of the point (0, b)Where a graph intersects the y-axis.

5.3.3b “The Slope of a Line”

The graphs of the equations y = 3x + 1 and y = x + 1 are drawn. Using the markedpoints on the graph, the ratio of the vertical distance to the horizontal distance between two points can be found.

y

x

y = x

+ 1

y =

3x

+ 1

2 (rise)

2 (run)

If you take the points (0, 1) and (-1, -2) on the line y = 3x + 1,the vertical distance is 3 units and the horizontal distance is 1 Unit or vertical distance (rise) = 3 or 3horizontal distance (run) 1

The equation y = 3x + 1 where 3/1 stands for the slope.

5.45.4 ““Rewriting the Linear Rewriting the Linear EquationEquation

Ax + By = C in the form Y = Ax + By = C in the form Y = Mx + BMx + B

And Vice Versa”And Vice Versa”•One convenient way so an equation canBe also be used to graph theSame line is to solve for Y In terms of X. When the equation Is transformed into y = mx + b

• The independent variable, and b arethe consonants.

The letter representing elements from the domainIs called the independent variable. For example,In Y= 3x-2, X is the independent variable.The letter representing elements from the range

•Is called the dependent variable. It’s valuedepends on X.

Any equation of the form Ax + By = CCan be transformed to an equivalent linearEquation Y= Mx+ B, which is also the Y-form.

•Illustrative examples

Simply the equation, by solving for yIn terms of x 3x + 4y = 12

Solution:

3x + 4y = 12 3x + (-3x) + 4y = (-3x) + 12 Addition property of equality 4y = -3x + 12 Additive inverse property(1/4) 4y = (-3x + 12) 1/4 Multiplication property of equality Y= -3/4 x + 3 Multiplicative inverse property

On the other hand, any linear equationOf the form y = mx + b can be transformedTo Ax + By = C

“Graph of a Linear Equation in two Variables”

The graph of a linear equation canBe drawn in the coordinate plane usingThe x- and y- intercepts of theLine, any two points on the lineOr the slope and a given point.

•The equation y = mx + b is known as theSlope-intercept form of the equation ofA line, where m is the slopeAnd b is the y-intercept. They interceptIs the point where the line intersectsThe y-axis and the x intercept is the

•Point where the line intersects the x-axis.To find the y-intercept in a givenEquation solve for y when x=0. Similarly,To find the x-intercept in a givenEquation, solve for x, when y=0.

•Illustrative ExampleDraw the graph of the equation

4x + 3y=12

y

x

4x + 3y =

12

0

-2

-2 2

2

-4

-4

4

4

Solution:

If y=0, 4x + 3(0) =12 4x = 12X= 4, y-intercept

With x-intercept 3, and y-intercept 4, the

Graph in the line that connects thePoints (3,0) and (0,4) is the

coordinatePlane shown above.

Increasing/Decreasing Increasing/Decreasing Graph of Graph of

y = mx + by = mx + bThe graph of the linear equation

may either be increasing or decreasing, depending upon the trend of the line.

ILLUSTRATIVE EXAMPLESA. Consider the graphs of y=3x-2 and y= -2x+3

4

2

-4 -2  0 2 4

-2

  -4

y=3x

-2

y=-2x+

3X

Y

What is the slope of y = -2x+3?What is the

slope of y=3x-2?

What is the relation of the

slope to the trend of the

line?

When the slope is positive, as inY=3x-2, the graph of the line is Increasing, or the line rises uniformly fromLeft to right . When the slope is Negative as in y=-2x+3, the graph of

The line is decreasing, or the line Falls uniformly from left to right.

““Obtaining the Obtaining the Equation of a Line”Equation of a Line”

The graph of Ax+By=C (A & B not both 0) is a

line in the coordinate plane.

Its basic characteristics have also been identified.

the geometric conditions used to describe any

given line in the coordinate plane will

be useful in finding the equation of

a line.

An equation for a line can be

obtained given:

1. the slope and one point on the line

2. two points on the line 3. the slope and its y-intercept

ILLUSTRATIVE EXAMPLES

A. The slope of a line is -2

and one point on the line is

(2,3). Find the equation of

the line.

4

2

-4 -2 0 2 4

-2

-4

(2,3)

(x,y)

X

Solution: let (x,y) be any point on the line other than (2,3). Using the slope formula, m=y-y1/x-x1 and replacing with the given values, -2=y-3/x-2.

Simplifying, -2 (x-2) = y-3 -2x + 4 = y-3 -2x –y =-7 or 2x +y= 7Check: does (2,3) satisfy the equation? 2 (2) + 3 = 7 4 + 3 = 7Therefore, 2x + y = 7 is the desired equation

Problem involving Problem involving linear equationslinear equationsSolution: The table shows the relationship between the perimeter of a square and its sides. Use the relation y= 4x

s (x) 2 4 6 8 10 12 14

P (y) 8 16 24 32 - - -

Many of the problems that are encountered

in daily life involve linear relations.

A.The perimeter of a square depends

upon the length of its side. Show how the perimeter changes as the length

of a side of the square changes.

Consider the number line below.

0 44 1 12 2 33

Measuring a distance of 4 units from the origin, two correct answers, -4 and +4 are obtained. Since no specific direction is given, count in either direction.

Remember:

The distance between 0 and any real number n on the number line is called the absolute value of the number. It is denoted by /n/, read as “absolute value of n”.

5.5.1 “Graph of Absolute Value “Using the corresponding table of values, the accompanying graphs are obtained.

y = x

x

y

2-2

2

-2

4

4

-4-4

x -3 -2 -1 0 1 2 3

y -3 -2 -1 0 1 2 3

y = x

Using the table of values for y = /x/, the graph of the basic absolute value function is drawn below. Unless otherwise specified, the domain of the function is the set of real numbers.

x

y

y = /x/X -3 -2 -1 0 1 2 3

y 3 2 1 0 1 2 3

The shape of the graph of the absolute value y = /x/ above reminds us that the value of every real number n is always nonnegative.