h.melikian/1100/041 graphs and graphing utilities(1.1) linear equations (1.2) formulas and...

H.Melikian/1100/04 1

Graphs and Graphing Utilities(1.1) Linear Equations (1.2)

Formulas and Applications(1.3)

Lect #4

Dr .Hayk MelikyanDepartmen of Mathematics and CS

[email protected]


-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1-2

-3

-4-5

Origin (0, 0)

Definitions: Rectangular coordinate system

The horizontal number line is the x-axis.

The vertical number line is the y-axis. The point of intersection of these axes is

their zero points, called the origin.


-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1-2

-3

-4-5

2nd quadrant 1st quadrant

3rd quadrant 4th quadrant

Definitions

The axes divide the plane into four quarters, called quadrants.

Each point in the rectangular coordinate system corresponds to an ordered pair of real numbers, (x, y).


Example

The first number in each pair, called the x-coordinate, denotes the distance and direction from the origin along the x-axis.

The second number in each pair, called the y-coordinate, denotes the distance and direction from the origin along the y-axis or along a line parallel to the y-axis.

Plot the point (3,2).Start at the origin and move 3 units to the right.From that point, move 2 units up.Now plot your point.


Graphs of equations

A relationship between two quantities can be expressed as an equation in two variables, such as

y = 2 + x2

A solution of an equation in two variables, x and y, is an ordered pair of real numbers with following property: when the x-coordinate is substituted for x and y-coordinate is substituted for y in the equation, we obtain a true statement.

The graph of an equation in two variables is the set of all points whose coordinates satisfy the equation. One method for graphing such equations is the point-plotting method.


Text Example

Sketch the graph of y = x2 – 4.

Let x = 3, then y = x2 – 4 = 9 – 4 = 5. The ordered pair (3, 5) is a solution to the

equation y = x2 – 4. We also say that (3, 5) satisfies the

equation.


(3, 5)y = 32 – 4 = 9 – 4 = 53

(2, 0)y = 22 – 4 = 4 – 4 = 02

(1, -3)y = 12 – 4 = 1 – 4 = -31

(0, -4)y = (0)2 – 4 = 0 – 4 = -40

(-1, -3)y = (-1)2 – 4 = 1 – 4 = -3-1

(-2, 0)y = (-2)2 – 4 = 4 – 4 = 0-2

(-3, 5)y = (-3)2 – 4 = 9 – 4 = 5-3

Ordered Pair (x, y)y = x2 – 4x

Text Example Cont.

• First, find several ordered pairs that are solutions to the equation.


-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1-2

-3

-4-5

Text Example Cont.

• Now, we plot these ordered pairs as points in the rectangular coordinate system.


Intercepts

An x-intercept of a graph is the x-coordinate of a point where the graph intersects the x-axis.

The y-coordinate corresponding to an x-intercept is always zero.

A y-intercept of an graph is the y-coordinate of a point where the graph intersects the y-axis.

The x-coordinate corresponding to an y-intercept is always zero


Example

Graph 4y + 5x = 20.Substitute zero for x:

4y = 20 or y = 5.Hence, the y-intercept is (0,5).Substitute zero for the y:

5x = 20 or x = 4.Hence, the x-intercept is (4,0).


Terms Involving Equations

3x - 1 = 2

An equation consists of two algebraic expressions joined by an equal sign.

3x – 1 = 2 3x = 3 x = 1 1 is a solution or root of the equation

Left Side Right Side

Linear Equations


Definition of a Linear Equation

• A linear equation in one variable x is an equation that can be written in the form

• ax + b = 0

• where a and b are real numbers and a = 0.


An equation can be transformed into an equivalent equation by one or more of the following operations.

Example1. Simplify an expression by removing grouping symbols and combining like terms.

3(x - 6) = 6x - x3x - 18 = 5x

-18 = 2x-9 = x

Divide both sides of the equation by 2.

3. Multiply (or divide) on both sides of the equation by the same nonzero quantity.

Subtract 3x from both sides of the equation.

3x - 18 = 5x3x - 18 - 3x = 5x - 3x-18 = 2x

2. Add (or subtract) the same real number or variable expression on both sides of the equation.

-9 = xx = -9

4. Interchange the two sides of the equation.

Generating Equivalent Equations


Solving a Linear Equation

• Simplify the algebraic expression on each side.

• Collect all the variable terms on one side and all the constant terms on the other side.

• Isolate the variable and solve.• Check the proposed solution in the

original equation.


Solve the equation: 2(x - 3) - 17 = 13 - 3(x + 2).

Solution

Step 1 Simplify the algebraic expression on each side.

2(x - 3) – 17 = 13 – 3(x + 2) This is the given equation.

2x – 6 – 17 = 13 – 3x – 6 Use the distributive property.

2x – 23 = - 3x + 7 Combine like terms.

Text Example



Solution

Step 2 Collect variable terms on one side and constant terms on the other side. We will collect variable terms on the left by adding 3x to both sides. We will collect the numbers on the right by adding 23 to both sides.

2x – 23 + 3x = - 3x + 7 + 3x Add 3x to both sides.

5x – 23 = 7 Simplify.

5x – 23 + 23 = 7 + 23 Add 23 to both sides.

5x = 30 Simplify.

Text Example



Solution

Step 3 Isolate the variable and solve. We isolate the variable by dividing both sides by 5.

5x = 305x/5 = 30/5 Divide both sides by 5

x = 6 Simplify.

Text Example



Solution

The solution set is {6}.

Step 4 Check the proposed solution in the original equation. Substitute 6 for x in the original equation.

2(x - 3) - 17 = 13 - 3(x + 2) This is the original equation.

-11 = -11 This true statement indicates that 6 is the solution.

2(6 - 3) - 17 = 13 - 3(6 + 2) Substitute 6 for x.?

2(3) - 17 = 13 - 3(8) Simplify inside parentheses.?

6 – 17 = 13 – 24 Multiply.?

Text Example


Types of Equations

• Identity:An equation that is true for all real numbers.

• Conditional: An equation that is true for at least one real number.

• Inconsistent: An equation that is not true for any real number.


Determine whether the equation 3(x - 1) = 3x + 5 is an identity, a conditional equation, or an inconsistent equation.

Solution To find out, solve the equation.

3(x – 1) = 3x + 5

3x – 3 = 3x + 5

-3 = 5

This equation is inconsistent.

Example


Text Example

Use the formula E = 0.177t + 71.35 to determine the year of birth for which U.S. women can expect to live 82 years.

First, substitute 82 for E in the formula: 82=0.177t + 71.35.

Next, solve for t by subtracting 71.35: 10.65=0.177t.

Now, divide both sides by 0.177.60.17 = t


Solution cont.

The formula indicates that U.S. women born approximately 60 years after 1950, or in 2010, can expect to live 82 years.


Strategy for Problem Solving

Step 1: Read the problem carefully. Attempt to state the problem in your own words and state what the problem is looking for. Let x (or any variable) represent one of the quantities in the problem.

Step 2: If necessary, write expressions for any other unknown quantities in the problem in terms of x.


Strategy for Problem Solving

Step 3: Form a verbal model of the problems conditions and then write an equation in x that translates the verbal model.

Step 4: Solve the equation and answer the question in the problem.

Step 5: Check the proposed solution in the original wording of the problem, not in the equation obtained from the words.


FORMULAS:

I = PRT Simple Interest=Principal *Rate * Time D = rt Distance = rate * time F = (9/5)C +32 degrees Fahrenheit = (9/5)degrees Celsius

+32


Example

Two consecutive numbers have a sum of 91. What are the numbers?


Solution

Step 1: Read the problem carefully. Attempt to state the problem in your own words and state what the problem is looking for. Let x (or any variable) represent one of the quantities in the problem.

Step 2: If necessary, write expressions for any other unknown quantities in the problem in terms of x.

Let x = The First Consecutive Number.Let x + 1 = The Second Consecutive Number


Solution cont.

Step 3: Form a verbal model of the problems conditions and then write an equation in x that translates the verbal model.

x + (x + 1) = 91


Solution cont.

Step 4: Solve the equation and answer the question in the problem.

x + (x + 1) = 91

2x + 1 = 91subtract 1 from each side -1 -1

2x = 90divide each side by 2 --- ---

2 2

x = 45


Solution cont.

Step 5: Check the proposed solution in the original wording of the problem, not in the equation obtained from the words.


Example

Solve the formula 4at+5y=2t+9y for t


Solution

4at + 5y = 2t + 9y4at + 5y - 2t = 9y bring all t terms to one

side

4at - 2t = 9y - 5yMove all other terms over

t(4a - 2) = 4y Factor out t

t = 4y / (4a-2) Divide by 4a-2

t = 2y/(2a - 1) Cancel a factor of 2


Formulas and Applications

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