Linear Equation and Quadratic Equation

8/4/2019 Linear Equation and Quadratic Equation

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Linear Equation andQuadratic Equation

Morales, Francisco Raphael

Sarmiento, Miguel Alfonso

Zapata, John Patrick


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Linear Equation

AN EQUATION is an algebraic statement in which the verbis "equals" = . An equation involves an unknown number,

typically calledx. Here is a simple example:

x+ 64 = 100.

"Some number, plus 64, equals 100."We say that an equation has two sides: the left side,x+

64, and the right side, 100.

In what is called a linear equation,xappears only to the

first power, as in the equation above. A linear equation is alsocalled an equation of the first degree. (The degree of any

equation is the highest exponent that appears on the

unknown number.)


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Now, the statement -- the equation -- will

become true only when the unknown has a certain value,

which is called the solution to the equation.

The solution to that equation is 36 -- which is easily foundby subtracting:

x= 100 64

x= 36.

36 is the only value for which the statement, "x+ 64 =100," will be true. We say thatx= 36 satisfies the equation.

Now, algebra depends on how things look. As far as how

things look, then, we will know that we have solved an

equation when we have isolatedxon the left.Why the left? Because that is how we read, from left to

right. "xequals . . ."
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In the standard form of a linear equation -- ax+ b = 0

--xappears on the left, not the right.

In fact, we are about to see that for any equationthat looks like this:

x+ a = b,

the solution will look like this:

x= ba.


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Inverse operations

There are two pairs of inverse

operations. Addition and subtraction,multiplication and division.

Formally, to solve an equation we must

isolate the unknown(typicallyx) on the left.axb + c = d.

To solve that equation, we must get a,

b, c over to the right, so thatxalone is on theleft.


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The question is:

How do we shift a number from one side of an

equationto the other?

Answer:

By writing it on the other side with

the inverse operation.For, that preserves the arithmetical relationship on the one

hand between addition and subtraction:

100 64 = 36 implies 100 = 36 + 64;

and on the other, between multiplication and division:10/2 = 5 implies 10 = 2.5

Algebra is, after all, abstracted -- drawn from -- arithmetic.


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And so, to solve this equation:

axb + c=d

then since b is subtractedon the left, we will addit on theright:

ax+ c=d+ b.

Since c is addedon the left, we will subtractit on the right:

ax=d+ bc.And finally, since amultiplies on the left, we will divide it on

the right:

x=d+ bc

a

We have solved the equation.


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The four forms of equations

Solving any linear equation, then, will fall into four

forms, corresponding to the four operations ofarithmetic. The following constitute the basic rules for

solving any linear equation. In each case, we will

shift a to the other side.

1. Ifx+ a = b, then x = b a."If a number is added on one side of an

equation,

we may subtract it on the other side."

2. Ifx a = b, then x = b + a.

"If a number is subtracted on one side of an

equation,

we may add it on the other side."


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3. If ax = b, then x = b

a.

"If a number multiplies one side of anequation,

we may divide it on the other side."

4. If x= b, then x = ab.

a"If a number divides one side of an equation,

we may multiply it on the other side."

In every case, we shifter a to the other side by

means of the inverseoperation. Every linear equation

can be solved by combining those four formal rules.
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Transposing

When the operations are addition or subtraction, that is

called transposing.

We may shift a term to the other side of an

equation by changing its sign.

+ agoes to the other side as a.

a goes to the other side as + a.

Transposing is one of the most characteristic operations of

algebra, and it is thought to be the meaning of the

word algebra, which is of Arabic origin. (Arabic mathematicians

learned algebra in India, from where they introduced it into

Europe.) Transposing is the technique of those who actually usealgebra in science and mathematics -- because it is skillful. And

as we are about to see, it maintains the clear, logical sequence

of statements. Moreover, it emphasizes that we do algebra with

our eyes.
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When we see

x+ a=b,

then we immediately see that +a goes to the otherside as a:

x=ba.

The way that is often taught these days, is to addato both sides, draw a line, and add:


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A logical sequence of statements

In an algebraic sentence, the verb is typically the equal sign = .

axb + c = d.

That sentence -- that statement -- will logically imply other

statements. Let us follow the logical sequence that leads to

the final statement, which is the solution.

(1) axb + c = d

implies (2) ax= d+ bc

implies (3) x= d+ bc .

a

The original equation (1) is "transformed" by first transposingthe terms. Statement (1) implies statement (2).

That statement is then transformed by dividing

by a. Statement (2) implies statement (3), which is the

solution.
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Thus we solve an equation by transforming it --

changing its form -- statement by statement, line by

line according to the rules of algebra, untilxfinally isisolated on the left. That is how books on

mathematics are written (but unfortunately not

books that teach algebra!). Each line is its own

readable statement that follows from the line above -- with no crossings out

In other words, What is a calculation? It is a

discrete transformation of symbols. In arithmetic we

transform "19 + 5" into "24". In algebra we

transform "x+ a = b" into "x= ba."
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Transposing versus exchanging sides

Example 1.a + b = cx

We can easily solve this -- in one line -- simply bytransposingxto the left, and what is on the left, to the right:

x = cab.

Example 2.a + b = c +x

In this Example, +xis on the right. Since we want +xon the left,we can achieve that by exchanging sides:

c +x= a + b

Note: When we exchange sides, no signs change.

The solution easily follows:c +x= a + bc

In summary, when xis on the right, it is skillful simply to

transpose it. But when +xis on the right, we may exchange the

sides.
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Canceling

If equal terms appear on both sides of an

equation,then we may "cancel" them.

x+ b + d = c + d.

dappears on both sides. Therefore, we may cancel

them.

x+ b = c.

Theoretically, we can say that we

subtracted dfrom both sides.

Finally, on solving forx:

x = cb.


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The unknown on both sides

Example 3. Solve forx:

4x 3=2x 11.

1. Transpose thex's to the left and the numbers to the right:4x 2x=11 + 3.

2. Collect like terms, and solve:

2x=8

x=4.This is another example of doing algebra with your eyes. You

should see that 2xgoes to the left as 2x, and that 3 goes to the right

as +3.

As a general rule for solving any linear equation, we can now

state the following:

Transpose all the terms that involve the unknown to the left, and add

them;

transpose the remaining terms to the right;

make 1 the final coefficient of the unknown, by dividing or multiplying.
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Simple fractional equations

Example 4. x = 4.

2

Since 2 divides on the left, it will multiply on the right:

x=2 4

=8.

Example 5. Solve forx:

4

x = 5.Solution. In the standard form of a simple fractional equation,xis in the

numerator. But we can easiy make that standard form by taking

thereciprocal of both sides.

x= 1

4 5This implies

x = 4 1

5

= 4

5.
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Example 6. Fractional coefficient.

3x=y

4

Since 4 divides on the left, it will multiply on the

right: 3x=4y. And since 3 multiplies on the left, it will divide

on the right:

x=4y

3

In other words, 3 goes to the other side as its reciprocal, 4

4 3.

Note that 3 is the coefficient ofx:

43x= 3

4 4x.

Coefficients go to the other side as their reciprocals!
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Linear Equation


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Let's draw the graph of this equation.

One method we could use is to find the x and y values of two

points that satisfy the equation, plot each point, and then

draw a line through the points. We can start with any two x

values we like, and then find y for each x by substituting the xvalues into the equation. Let's start with x = 1.

Value of x y =1/2 x + 2 Value of y

1 y = . 1 + 2 = + 2 2.5

2 y = . 2 + 2 = + 2 3


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Let's plot these points and draw a line through them.


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Graphing Using Slope and Y-Intercept

There's another way to graph an equation using your

knowledge of slope and y-intercept. Look at the equation

again.We can find the slope and y-intercept of the line just by

looking at the equation: m = 1/2 and y intercept = 2.

Just by looking at these values, we already know one point on

the line! The y-intercept gives us the point where the lineintersects the y-axis, so we know the coordinates of that point

are (0, 2), since the x value of any point that lies on the y axis

is zero.


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To find the second point, we can use the slope

of the line. The slope is , which gives us the

change in the y value over the change in the x

value. The change in the x value, the

denominator, is 2, so we move to the right 2

units.

The change in the y value, the numerator, is

positive one. We move up one unit. This gives

us the second point we need. Now we candraw the line through the points.


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Examples:


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Quadratic Equation

A QUADRATIC is a polynomial whose highest exponent is 2.ax + bx+ c.

Question 1. What is the standard form of a quadratic equation?

ax + bx+ c= 0

The quadratic is on the left. 0 is on the right.

Question 2. What do we mean by a rootof a quadratic?

A solution to the quadratic equation.

For example, the roots of this quadratic

x + 2x 8are 4 and 2. For, we can factor that quadratic as

(x+ 4)(x 2).
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Now, if x= 4, then the first factor will be 0. While if x= 2, thesecond factor will be 0. But if any factor is 0, then the entireproduct will be 0. That is, if x= 4 or 2, then

x + 2x 8 = 0.

Therefore, 4 and 2 are the solutions to the quadraticequation. They are the roots of that quadratic.

Question 3. How many roots has a quadratic?

Always two. Because a quadratic (with leading coefficient1, at least) can always be factored as (xa)(xb),and a,bare the two roots.

Note that if a factor is (x+ q), then the root is q. For,(x+ q) can take the form (xa):(x+ q) = [x (q)].qis the root,
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Question 4. What do we mean by a double root?

The two roots are equal. That is, the factors are

(xa)(xa), so that the two roots are a, a.

For example, this quadratic

x 10x+ 25

can be factored as

(x 5)(x 5).

Ifx= 5, then each factor will be 0, and therefore the

quadratic will be 0. 5 is called a double root.

When will a quadratic have a double root? When thequadratic is a perfect square trinomial.
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The quadratic formula

Here is the quadratic formula -- which is proved by

completing the square In other words, the quadratic formula

completes the square for us.

Theorem. If

ax + bx+ c = 0,

then

We will prove this below.

Example 4. Use the quadratic formula to solve this quadraticequation:

3x + 5x 8 = 0
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Linear Equation and Quadratic Equation

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