Thursday, September 21, 2023

# Standard Form Definition Math Algebra

## What Is The Standard Form Of A Linear Equation

Standard form for linear equations | Algebra I | Khan Academy

The standard form of a linear equation, also known as the general form, is:

 Standard Form :ax+by=c

The letters a, b, and c are all coefficients. When using standard form, a, b, and c are all replaced with real numbers. The letter x represents the independent variable and the letter y represents the dependent variable.

A few notes on Standard Form:

• The a term must be a positive integer
• a, b, and c cannot be decimals or fractions

## Standard Form Of Polynomial

The standard form polynomial is written with the exponents in decreasing order to make calculations easier. A polynomial is said to be in its standard form, if it is expressed in such a way that the term with the highest degree is placed first, followed by the term which has the next highest degree, and so on. There are two very simple rules of writing a polynomial in a standard form, they are:

• Write the terms in the descending order of their powers .
• Ensure the polynomial contains no like terms.
• Hence, the standard form is anxn + an-1xn-1 + an-2xn-2 + … + a1x1 + a0. For example, the standard form of equation y2 + 7y6 – 8y – y9 is written as -y9 + 7y6 + y2 – 8y.

Note: The thumb rule for writing a polynomial in its standard form is D-U. D stands for Descending and U stands for unlike terms.

## Converting Numbers Into Standard Form

Numbers in standard form are written as a multiple of a power of 10. In the case of large numbers, the power of 10 will be large, meaning a positive exponent. For small numbers, the power of 10 will be extremely small , meaning a negative exponent.

In order to convert a number into standard form, follow these steps:

• Move the decimal point until there is only one non-zero digit to the left of the decimal point. The number that has been formed is the value for A. For example, 5000 becomes 5.000, and we can remove the leading 0’s giving us 5.
• Count the number of times that the decimal point was moved. If the decimal point was moved to the left, the value for n in the formula will be positive. If the decimal point was moved to the right, the value for n in the formula will be negative. In the case of 5000, the decimal point was moved to the left 3 times, meaning n is equal to 3.
• Write the number in the form using your results from step 1 and step 2.
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## Standard Form Of A Linear Equation

The standard form of linear equations also known as the general form is a method of writing linear equations. A linear equation can be written in different forms like the standard form, the slope-intercept form, and the point-slope form. The standard form of a linear equation is expressed in two ways, with one variable and with two variables. The standard form of linear equation with one variable is expressed as ax + b = 0 where a and b are constants and the letters x is the variable. The standard form of a linear equation with two variables is expressed as ax + by = c. where a, b, and c are real numbers and the letters x and y are the variables. Look at the image below.

Lets see how to convert the two lines into the standard form of a linear equation ax + by = c.

Line 1: x + y = 7 i.e. 1x +1y = 7. Here, a = 1, b =1 , c = 7

Line 2: y = 3x i.e. 3x -1y = 0. Here, a = 3, b = -1 , c = 0

Therefore, what we see here is the standard form of equation which is linear i.e. ax + by = c.

## Examples Of Standard Form Of Fractions

• The most famous standard fraction is 22/7, also called pi.
• All unit fractions, i.e., fractions with numerator 1 are standard.

The definition for the standard form of decimal numbers is the same as that of whole numbers,

Any number that we can write as a decimal number, between 1.0 and 10.0, multiplied by a power of 10, is said to be in standard form.

The scoop about the standard form of decimal numbers lies in their steps. This is a bit different from whole numbers, where you may have noticed, the power of 10 was a positive number like in 1.23 × 108.Lets see how we can convert a scary-looking decimal number into a pretty-looking decimal number.

Consider the decimal number 0.0004789.

Step 1: Make your decimal number jump from its original position and place it after the first non-zero digit: 4.789.

Step 2: Count the number of digits you jumped. In this case, we jumped 4 digits. This number will be the power of 10 while writing our decimal number in standard form.

Step 3: If you have made jumps in the direction to the right, the power of 10 will be negative. If you made the jumps in the direction to the left, the power of 10 will be positive.

Step 4: So, in standard form: 0.0004789 is 4.789 × 10-4.

Using the same method, the standard form of the decimal number 981.23 will be

9.8123 × 102.

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## System Of Equations With Standard Form

Lets see a quick example. If we were given the system of equations:

y=-4x+9

y-9=\fracx-4

we can rewrite the equations in standard form.

y+4x=9

Then, we can solve using the elimination method by multiplying the second equation by 4.

y+4x=9

8y-4x=40

Then, we can substitute the value of y into one of the original equations to determine the value of x.

2y-x=10

2-x=10

-x=10

-x=10-

x=-10+

x=\frac

Now we have solved the system of equations. The solution is . Using standard form allowed us to use the elimination method to solve the system.

As well see below, standard form is also useful for easily determining the intercepts of a linear function.

## Writing The Statements In Algebraic Form

• The total cost of 1 kg apple and 2 kg of guava is \$3. If x represents the cost of 1 kg apple and y represents the cost of 1 kg guava, then this statement is represented in the form of an algebraic expression as:
• x + 2y = 3

• If the area of the rectangle is 36 cm2. Its length is represented as l and width is represented as b. Then this statement is represented as :
• l ×b = 36

## How To Write Numbers In Standard Form

When learning about numbers, well be introduced to the numbers standard and expanded form. The numbers were working with are normally in standard form but we can also write numbers in expanded to standard form by simplifying them.

When given the expanded form of a number, follow the steps below to simplify them and write them in standard form:

• Take the expanded forms sum vertically or horizontally. Here are examples of what we meant by simplifying the expanded form to its standard form:
• ## Standard Form Of Equations

Converting from slope-intercept to standard form | Algebra I | Khan Academy

The standard form of an equation is where zero goes on the right and everything else goes on the left. i.e., it is of the form

This helps in solving the equation in a simple manner. Equations used for polynomials, linear and quadratic have a standard form, let us look at what they are.

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## How To Write Linear Equations In Standard Form

We can write the line equations standard form by isolating the linear equations constant on the right-hand side of the equation. Heres the standard form or also known as the general form of the linear equation:

The process is straightforward isolate all the terms on the left-hand side of the equation. Apply appropriate algebraic techniques when needed. Of course, the best way to master the process of writing linear equations in standard form is through practice. When youre ready, try out the sample problem below!

## What Is The Standard Form Of A Number

A standard form of a number in Maths is basically mentioned for the representation of large numbers or small numbers. We use exponents to represent such numbers in standard form.

The correct definition of standard form could be explained better in terms of decimal numbers and following certain rules. As we know, are the simplified form of fractions.

Some fractions give decimal numbers which have numbers after decimal at thousandths, hundredths or tenths place. But there are some fractions, which give a big decimal number. To represent such big numbers, we use simpler forms, which are also stated as Scientific notation.

Basically, we can say, it is the representation of rational numbers in standard form. Rational numbers are numbers that can be represented in the form of p/q, where p and q are both integers. For example, 1/10, 4/5, 8/9, etc.

## What Is Standard Form

A standard form is a form of writing a given mathematical concept like an equation, number, or an expression in a form that follows certain rules. Representing very large or very small numbers concisely, the standard form is used. For example, 4.5 billion years is written as 4,500,000,000 years. As you can see here, writing a large number like 4.5 billion in its number form is not just ambiguous but also time-consuming and there are chances that we may write a few 0s less or more while writing in the number form. In this case, writing the number in standard form is very helpful. For example, the standard form of 4,500,000,000 = 4.5 × 109. Not only numbers but the fractions, equations, expressions, polynomials, etc also can be written in the standard form.

Let us study the standard form of each of these in detail.

## Standard Form Definition Algebra

Definition of Standard Form. All of the things you see in maths, like numbers or fractions or equations or expressions, have a standard form defined for them. We can think of the standard form as the most common way of representing a mathematical element. Lets look at the standard form of some commo See more

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## Tips To Master Standard Form

• Thoroughly understand the two parts of the standard form: digits and the power of 10.
• Remember that the power is negative if you have made jumps to the right and the power is positive when you have made jumps to the left while putting a decimal number in its standard form.
• Count the places you have moved the decimal point twice before writing the final answer.
• Dont forget about the second cousin of prime numbers: the co-primes. They are the basics of putting a fraction in its standard form.

## How To Write A Linear Equation In Standard Form

Lets write an equation of the line with a slope of 4 and a y-intercept of 7 in standard form.

To begin, we will first write the equation in slope-intercept form. This is the easiest form to write when given the slope and the y-intercept.

 Slope-Intercept Form:

We know the slope, m, is 4 and the y-intercept, b, is 7.

y=4x+7

To change this into standard form, all we need to do is subtract the x term from both sides, in this case 4x.

y-4x=7

We have now written the standard from of a linear equation. The linear equation with a slope of 4 and a y-intercept of 7 is y-4x=7.

Therefore, the y-intercept is at 10. This means the point is on the graph.

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## How To Write Higher

When writing quadratic equations or equations with higher degrees in standard form, we simply isolate all the terms on the left-hand side of the equation. This means that higher-degree equations standard forms will always have 0 on their right-hand side.

The main idea is that when writing equations other than linear equations in standard form, we simply move all the terms on the left-hand side so we have 0 on the right-hand side of the equation. Of course, for some cases, well need to apply appropriate algebraic techniques to simplify the given equation.

## Converting To Standard Form

Writing a number in standard form | Arithmetic properties | Pre-Algebra | Khan Academy

Different forms of linear equations are useful for different applications. For example, when solving systems of linear equations, it is helpful to first convert the equation into standard form. Slope-intercept and point-slope form are two commonly used forms of a linear equation that, while useful for graphing, are not useful for solving systems of linear equations. It is therefore important to know how to convert between various forms of linear equations to best suit the application.

Converting from point-slope or slope-intercept form to standard form involves moving all the variables to one side of the equation, moving the constant to the other side, then manipulating the equation as necessary such that the coefficients of the terms of the equation are integers.

Examples

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## Standard Form Of Slope Of A Line

The slope of a line is defined as the change in y coordinate with respect to the change in x coordinate of that line. Representing a line geometrically, we use the standard form of a linear equation . To determine the slope of a line that is expressed graphically, the equation must be converted to a slope-intercept form. To do this, we must solve the equation for y, and the standard form of a slope is expressed as y = mx + c, where m is the slope of the line. This formula is used when the line is straight.

When there are two points in a plane, the slope can be defined as the ratio of change in the value of y to the change in the value of x. The standard form of slope of a line is expressed as m = /. The image below represents both the coordinates on a graph.

## What Is The Definition Of Standard Form In Math

Standard form in math is the method of representing a particular element in the most common way. Very large numbers or very small numbers are expressed in the standard form. Mathematical elements such as equations are expressed in a standard form to better solve the problem. In other words, a standard form is a form of writing a given mathematical concept like an equation, number, or expression in a form that follows certain rules.

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## How To Write The Standard Form Of A Number

The steps to write the standard form of a number are as follows:

Step 1: Write the first number from the given number.

Step 2: Add the decimal point after the first number.

Step 3: Now, count the number of digits after the first number from the given number and write it in the power of 10.

For example, 52300000000 is the number. Thus the standard form of a number is obtained as follows:

Step 1: The first number is 5

Step 2: Adding the decimal point after 5, it becomes 5.

Step 3: The number of digits after 5 is 10.

Hence, the standard form of 52300000000 is 5.23× 1010.

## Standard Form Of Fraction

In the case of fractions, we need to ensure that in the standard form of fractions, the numerator and denominator must be co-prime numbers. That means they have no common factor other than 1, hence the standard form is also called the simplest form of a fraction. For example, 14/22 and 13/6. The simplest form of 14/22 = 7/22 and 13/6 is already in its simplest form as 13 and 6 are co-prime.

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## How To Express Large Numbers In Standard Form

Any number can be expressed as decimal numbers between 1 and 10 multiplied by a power of 10. This form of representation is termed as the standard form. We can understand this concept with the following examples.

500 = 5 × 102

567 = 5.67 × 102

56.78 = 5.678 × 101

The advancement in science and technology has introduced us to numbers as big as the diameter of the Earth and as small as the size of a human cell. In order to represent these numbers, we use the exponential form to make their reading and writing more convenient.

## Definition Of Standard Form

Did you know that Earth is 4,543,000,000 years old? We know you skipped past all those zeroes, so we will give you a simpler way of saying it: Earth is 4.543 billion years old. Do you see how reading numbers a certain way makes them easier to understand? Thats why all the mathematicians in the world decided to agree on some rules on writing mathematical concepts so that it is convenient for everyone to read, write, and work with. This particular way is called the standard form.

All of the things you see in maths, like numbers or fractions or equations or expressions, have a standard form defined for them. We can think of the standard form as the most common way of representing a mathematical element.

Lets look at the standard form of some common mathematical elements:

Whole Numbers

You can define the standard form of a whole number as follows.

Any number that we can write as a decimal number, between 1.0 and 10.0, multiplied by a power of 10, is said to be in standard form.

For example, take the number 123,000,000 an easier way of writing this number is

1.23 × 108

If you observe carefully, 1.23 is a decimal number between 1.0 and 10.0 and so we have the standard form of 123,000,000 as 1.23 × 108.