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Algebra 1

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Algebra 1 is a branch of mathematics in which we represent the components of a specified set or the numbers by use of symbols, alphabetical letters to express general relationship between the components of set.  Algebra is a way of simplifying Math through the use of variables, formulas or equations.

Algebra 1 is a broad topic which covers several topics like solving rational equations, algebraic equations, linear and quadratic equations etc.

Real Numbers: 

This topic contains Rational Numbers on number line, addition $(+)$, subtraction $(-)$ multiplication $(\times)$, division of rational numbers, square root $(\sqrt{x})$ and Cube root $(x^{\frac{1}{3}})$ of Real Numbers.

Solution of Linear Equations:

This topic contains solution of various linear equations by using property of addition $(2x + 3x = 5x)$, subtraction $(5x - 2x = 3x)$, multiplication $(7x \times  8x = 56 x^2)$ and division ($\frac{12x}{4x}$ = $3$) with variable on each side. Also, the properties of ratio $(a : b)$ and proportion $(a : b :: c : d)$, percentage $( \%)$ come under this branch.
Factoring:

This topic contains greatest common factor, factorizing of trinomials $(ax^{2} + bx + c)$ by splitting middle term or by the method of perfect square.

Statistics:

It is another branch of algebra 1 which includes introductory matrices, measure of variation, making of histograms, sampling and bias.

Probability:

Probability contains possibility of occurring of an event from a group of events under (distributional, conditional, binomial etc.). It also contains Permutation $(^{n}P_{r})$ and Combination $(^{n}C_{r})$.
Solving System of Linear Equation and Inequalities:

Linear equations of one and two variables, and the method of solving them by substitution, elimination, completing square method, solving by graphical method comes under the branch of algebra 1.

Algebra 1 Topics

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Algebra is a vast branch of mathematics. There are several topics that comes under Algebra 1 and here are few of them:
Polynomials:

In simple words, polynomial means, 'many terms'. They are the expression which include monomials, binomials, trinomial and also the higher order of polynomials. There are different types of polynomials and they are classified on the bases of number of the terms and the degree. Given below are the different types of Polynomials:

1) Monic Polynomial:
    If an expression is having a leading co-efficient of $1$, then such expressions are called as monic polynomials.

2) Quadratic Polynomial:
    The general expression of quadratic polynomial is $ax^{2} + bx + c, a  \neq 0$

3) Cubic Polynomial:
    The general expression of cubic expression of cubic polynomial is $ax^{3} + bx^{2} + cx + d, a \neq 0$

4) Prime Polynomial:
    A polynomial which cannot be factorized into the product of the two polynomials is called as prime polynomial.
    Ex: $x^{2} + x + 1$
Radical Expression

An expression containing a square root is called as radical expression. These expressions can be easily solved by performing squaring operation. 

Rational Expression
An expression where the numerator and denominator or both of them are polynomials are called as rational expressions. By simplifying the rational expression, we can reduce the expression into the lowest form.
Analyzing Linear Equations

A linear equation is an equation of a line, which contains two different variables and is easily transferred into graphical representation. The general form of linear equation is $ax + by + c$ = $0$, where $x$ and $y$ are variables, $a$ and $b$ are co-efficients of $x$ and $y$ and $c$ is constant.

Solving Linear Inequalities

Linear inequality is the comparison of two values. We can add, subtract, multiply and divide the inequalities to solve the linear inequalities. The concept of solving the linear inequalities is the same as solving the linear equation. Linear inequality has its own properties and they are as given below:

1) Transitive Property
    When $a, b$ and $c$ are real numbers, then
    If $a < b$ and $b < c$, then $a < c$

2) Addition Property:
    If $a < b$, then $a + c < b + c$

3) Subtraction Property:
    If $a < b$, then $a - c < b - c$

4) Multiplication Property:
If $a < b$ and $c$ is positive $c  \times  a < c  \times b$
If $a < b$ and $c$ is negative - $c \times a > - c \times b$
Quadratic Equation

A polynomial with a second degree is called as quadratic equation. When the discriminant is zero, then such quadratic equation is called as perfect square, i.e. $b^{2} -  4ac$ = $0$. When a discriminant is greater than $1$ then the roots of the equation are real and if it is lesser than $1$, at that point the roots of the equation are imaginary. The general form of quadratic equation is $ax^{2} + bx + c$ = $0$. And, the formula of quadratic equation is as follows:

$x$ = $\frac{-b\ \pm\ \sqrt{b^{2}\ -\ 4ac}}{2a}$
Simultaneous Equation

A set of two equations with two unknown variables are called as simultaneous equations. Solving simultaneous equations involves finding the values of the variables that will satisfy the equation. 

There are two methods of solving the simultaneous equations:

1) Substitution Method:
    Here, we first solve one of the equation for one of the unknowns and then, substitute the answer into the other equation.

2) Elimination Method:
    In this method, first make the co-efficient of one of the variables to the same value in both the equations. Then, add or subtract an equation     from the other to form a new equation that contains one variable.

Algebra 1 Problems

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Given below are few solved problems in Algebra 1.

Solved Examples

Question 1: Subtract 2x+ x + 1 from 16x+ 4x + 1
Solution:
Given: 16x+ 4x + 1 and 2x+ x + 1

Here first, let us find the opposite of  the subtracted term

That is,  2x+ x + 1 = - 2x- x - 1

By subtracting the two equations, we get

16x+ 4x + 1 - 2x- x - 1

= (16x- 2x3) + (4x - x) + 1 - 1

= 14x+ 3x

Question 2: Solve the radical expression $\sqrt{5y - 7}$ = y - 1
Solution:
Given: $\sqrt{5y - 7}$ = y - 1

First, we need to square both the sides

$\left [ \sqrt{5y - 7} \right ]^2$ = $(y - 1)^2$

By simplifying the above equation, we get 

5y - 7 = y- 2y + 1

After converting the above equation into factor form, we get

y2 - 7y + 8 = 0

We got the above equation in the quadratic equation. Let us solve this equation using the quadratic equation

x = $\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

Here, a = 1, b = -7, c = 8

$\therefore$ x $\frac{7\pm \sqrt{7^{2}-4.1.8}}{2}$

x = $\frac{7\pm \sqrt{49 - 32}}{2}$

x =  $\frac{7\pm \sqrt{17}}{2}$

x1 = $\frac{7+ \sqrt{17}}{2}$ and x$\frac{7- \sqrt{17}}{2}$

Question 3: Solve the linear inequalities 4x + 28 > 0
Solution:
Given: 4x + 28 > 0

First, subtract 28 on both the sides.

4x + 28 - 28 > 0 - 28

4x + 0 > 0 - 28

4x > -28

Now, divide 4 on both sides,

$\frac{4x}{4}$ > $\frac{-28}{4}$

x > -7

Question 4: Solve the following simultaneous equations using the elimination method
6x + 7y = 12
4x - 7y = 8
Solution:
Given: 
6x + 7y = 12 --> 1
4x - 7y = 8 --> 2

Since the co-efficient of one of the variables is numerically equal and also opposite signs, we can add 2 equations for eliminating y variable.

By adding the like term, we get

(6x + 4x) + (7y - 7y) = 12 + 8

That is, 10x = 20

Now, let us divide by 10 both the sides

$\frac{10x}{10}$ = $\frac{20}{10}$

$\therefore$ x = 2 

Now, substitute the value of x in the first equation

6(2) + 7y = 12

12 + 7y = 12

12 + 7y -12 = 12 - 12

7y = $\frac{0}{7}$

y = 0

=> x = 2,  y = 0