Category: Maths Notes PPT

The interactive unit circle is what that joins the trigonometric functions – sine cosine, and tangent, and the unit circle. The unit circle is actually referred to as a circle of radius one suspended in a specific quadrant of the coordinate system. The radius of a unit circle can be taken at any point on the perimeter of the circle.

It forms a right-angled triangle. The angle between this interactive unit circle will be displayed by angle θ. In order to change a grade, you would simply need to click and drag the two control points.

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Functions of Interactive Unit Circle

This unit circle basically consists of 3 functions as follows:

The interaction between this unit circle and its correlating functions is referred to as interactive unit circles.

Sine, Cosine and Tangent

1. Sine

The second and another basic trigonometric function is sine represented by θ. In Mathematical terms, sine θ is computed by dividing the perpendicular of a right-angled triangle by its hypotenuse. Thus, we can compute the length of the sides or the angle of any structure with the help of the above relation. Hence, the formula to calculate Sineθ is as below;

Sine θ = Perpendicular/Hypotenuse

• Cosecant: With respect to cosine, the reciprocal of sineθ is referred to as cosecant θ. It is computed by reciprocating sine or just by dividing it with 1. Hence, Cosecant θ = 1/sin θ.

2. Cosine

In a right-angled triangle, the ratio between the base and hypotenuse of a triangle is referred to as cosineθ. It is actually one of the most crucial trigonometric functions of all. In Mathematical terms, cosine is obtained by dividing the base of a right-angled triangle with its hypotenuse. Hence, formula to calculate Cosineθ is as below;

Cosine = Base/Hyp

• Secant: The reciprocal of cosine which is known as secant θ is also used in some triangles. The secant θ is used in several numerical calculations and is calculated by reciprocating cosine θ. Thus, Secant = 1/cosine.

3. Tangent

Another and 3rd basic trigonometric function is referred to as tangent. As per sine θ and cosine θ, we can also calculate and get the answer for tangent in a right-angled triangle. In a right triangle, the perpendicular of a triangle is divided with its base, and we easily obtain the value of tangent θ.

The mathematical formula to calculate tangent θ is: Tang θ = Perp/Base.

Thus, all the equations and the trigonometric functions can be understood by the interactive unit circle graph.

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Trigonometric Circle Interactive Simulation

Choose a Quadrant and drag the point in the simulation as shown in the figure to visualise the unit circle in all the four quadrants.

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Solved Examples

Example:

Why is cot 180° undefined?

Solution:

We know that 

• cot θ = 1/tan θ

• tan θ = sin θ/cos θ

∴ cot θ = cos θ/sin θ

From the interactive unit circle graph chart, we are familiar with:

sin180° = 0

Since, division by 0 is ∞, cot 180° = ∞

Hence, cot 180° = ∞

Example:

Calculate the exact value of tan 210° using the interactive unit circle.

Solution:

We are familiar with:

tan210° = sin210° / cos210°

Making use of the unit circle chart:

• sin 210° =  -1/2

• cos 210° = -√3/2

Therefore,

tan 210°=sin 210°/cos 210°

=−1/2 / −√3/2

=1/√3

=√3/3

Therefore, tan210° = √3/3

Key Facts

• The unit circle is referred to as a circle of radius 1 unit.

• The equation of a unit circle is x² + y² = 1.

• You can refer to the conversion table of angular measures to radian measures for finding important Sin Cos and Tan values of the 1st quadrant.

In general inverse meaning is “something that is opposite or reverses”. When we define inverse meaning we will talk mainly from the Mathematics point of view where we find the inverse of various operations and functions.

What is the Meaning of Inverse in Mathematics?

The inverse meaning in Math is “a function or operation which reverses the order or operation of another function or operation”.

Inverse Math Example: The inverse operation of addition is subtraction, the inverse operation of multiplication is division.

Now let us look into basic concepts of various operations and their inverse operation to understand the inverse Math definition clearly.

The Inverse of a Number

Consider a number x where x is not equal to zero. Then the inverse of a number x is the reciprocal of the number that is 1/x.

Ex: Inverse of a number 100 is 1/100, the inverse of a number 34 is 1/34.

The Inverse of Various Operations

In this section, we will clear our doubts about inverse meaning by applying the inverse to the basic operations of Mathematics.

1. Inverse Operation in Addition

The addition is one of the foremost operations in arithmetic where we add the numbers to find the total or sum of the numbers.

So, the inverse of the addition operation is subtraction. That is whenever we are asked to find the inverse of addition we have to subtract.

Ex: The addition of two numbers 25 and 15 can be written as 25 + 15 = 40. So, the inverse of this addition operation will be 40 – 15 = 25.

2. Inverse Operation on Subtraction

Subtraction is an operation where we remove numbers from the collection.

The inverse of the subtraction operation is addition.

Ex: The subtraction of two numbers 40 and 30 can be written as 40-30 = 10. So, the inverse operation of this will be 10 + 30 = 40.

3. Inverse Operation on Multiplication

Multiplication is an operation where we combine groups of equal sizes. Multiplication is nothing but a repeated addition process.

So, the division is the inverse of multiplication.

Ex: The multiplication of two numbers 4 and 6 is 4 × 6 = 24. The inverse operation of this is a division which is as follows: 24/6 = 4.

4. Inverse Operation on Division

The division is an operation where we divide the group of things into parts.

The inverse operation of division is multiplication.

Ex: The division of a number 45 by 5 is 45/5 = 9. The inverse operation of this is multiplication which is as follows: 9 × 5 = 45.

Additive Inverse

The Additive inverse of a number is the value that results in a sum as zero when added with the original value.

Ex: The additive inverse of +7 is -7. So, the sum will be +7 – 7 = 0.

Multiplicative Inverse

The Multiplicative inverse of a number is the value that results as one when multiplied with the original value.

Ex: The multiplicative inverse of 6 is 1/6. So, when we multiply these 2 values we get (6 × 1)/6 = 1.

The Inverse of a Function

The inverse function is a function that reverses the other function’s action.

Ex: Consider the function f(x) = 7x + 2 = y. So, the inverse function will be g(x) = (y – 2)/7 = x.

So, the inverse function of 7x + 2 is (y – 2)/7.

The Inverse of a Trigonometric Function

The trigonometric functions are the functions that relate the right-angled triangle angle with the ratios of the side of the triangle.

Ex: sin θ = Opposite side/Hypotenuse. So, we find the angle of the triangle by using this formula. What if we have to calculate the hypotenuse of a triangle if an angle is given. Then we will use the inverse function to calculate the hypotenuse. So, the inverse is written as

θ = sin-1(Opposite side/Hypotenuse) is the inverse of the trigonometric sine function.

Similarly, cos θ = Base/Hypotenuse. So, inverse is θ = cos-1(Base/Hypotenuse).

Tan θ = Opposite side/ Base. So, inverse is θ = tan-1 (Opposite side/ Base).

The Inverse of an Exponential Function

The inverse of an exponential function is a logarithmic function.

Ex: Consider an exponential function 43 = 64. So, the inverse of this function will be a logarithmic function log4 (64) = 3.

Problems on Inverse

1. Find the inverse of a number 4, 14, 25 and 36.

Ans: To find the inverse of a number, we have to take the reciprocal of the given numbers.

So, the inverse of a number 4 will be 1/4.

The inverse of a number 14 will be 1/14.

The inverse of a number 25 will be 1/25.

The inverse of a number 36 will be 1/36.

2. Find the inverse of the addition of 2 numbers 35 and 74.

Ans: The inversion of addition is subtraction. The sum of the numbers 35+74 = 109. So the inverse will be 109 – 74 = 35.

3. Find the inverse of the multiplication of the two numbers 5 and 9.

Ans: The inverse of the multiplication is division. So the product of the two numbers is 5 × 9 = 45. Therefore the inverse of the two numbers is 45/9 = 5.

4. Find the additive inverse of the following numbers. -5, -2, 5, 10.

Ans: The additive inverse 0f -5 is +5 because -5+5 = 0 which proves the additive inverse property.

The additive inverse of -2 is +2 because -2+2 = 0.

The additive inverse of 5 is -5 because +5-5 = 0.

The additive inverse of 10 is -10 because +10-10 = 0.

5. Find the inverse of the function 10y – 3.

Ans: To find the inverse of the given function consider f(y) = 10y – 3 = x. So, the inverse of this function will be (x + 3)/10.

6. Find the inverse of the exponential function 63.

Ans: We know that 63 = 216. So, the inverse of this exponential function is log6 216 = 3.

Conclusion

• The inverse operation of solving equations is important because it allows the reversal of Mathematical operations. One of the most important questions, once the Mathematical procedure is introduced, is how to reverse it.

• A function can be viewed as mapping things of one type to things of a different type. The opposite of the function indicates how the original value is returned. Without really thinking about that, we do a lot of inverse operations in everyday life.

Prime numbers are the numbers that have only two factors, the number itself and 1. 

To check if a number is prime or not we have to factorize it. So let us study whether 91 is a prime or composite number or why is 91, not a prime number? After factorizing 91, we get 4 factors. 91 has more than 2 factors i.e 1, 7,13, and 91 so 91 is not a prime number. In this article we will be studying what are prime numbers, how to determine a number, is prime or not, and is 91 a prime number?

What is a Composite Number?

Numbers containing more than two elements are called composite numbers. The number of components that a number has can be used to classify it. A prime number is one that contains just two factors: one and the number itself. Most numbers, however, include more than two elements and are referred to as composite numbers. We’ll look at the distinction between prime and composite numbers, as well as the lowest composite number and odd composite numbers, on this page. The last one is intriguing because, unlike 2, which is the only even prime number, there exist multiple odd composite numbers.

What is the Prime Number?

A prime number is an integer greater than one and can be divisible by only itself and one i.e it has only two factors. Zero, 1, and numbers less than 1 are not considered prime numbers.

A number having more than two factors is referred to as a composite number. The smallest prime number is 2 because it is divisible by itself and 1 only.

Steps to Check Prime Number

To check whether any number is a prime number or not first we have to take factors of that number. If it has only two factors that are the number itself and 1 then it is a prime number. The following are the steps to check whether the given number is prime or not.

To check whether the number is a prime number, first, divide the number by 2 if it is completely divisible by 2 it is not a prime number but if it is not divisible by 2, try with 3. Take the square root of a number and check the division test till the number is less than the square root. If it is not divisible by any number then it is a not prime number or else it is a prime number. So let us check if 91 is a prime or composite number?

Now let us follow this step for 91 and see if 91 is a prime number or not. 

Properties of a Prime Number

A prime number is a positive integer with two components that are precisely the same. If p is a prime, then the only factors it can have are 1 and p. A composite number is one that does not follow this pattern and maybe factored into other positive integers. A positive number or integer that is not a product of any other two positive integers other than 1 and the number itself is another method of describing it.

It’s worth noting that 1 is a non-prime number. It’s a one-of-a-kind number.

List of prime numbers to 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

91 a Prime Number or Not

To find whether 91 is prime or composite, first, check whether (9+1) is divisible by 2 if not check it by 3. 

We have, 9+1 is 10 which is not divisible by 3, continue with the following steps.

• Step 1: First take the square root of 91, 91−−[sqrt{}] = 9.53

• Step 2: Choose prime numbers less than 9 and see if any of the numbers divide 91 completely.

• Step 3: we get that the number 7 divides 91 perfectly. So, 91 is divisible by 7 we get more than two factors other than 1 and 91 itself. Hence, we can say that 91 is not a prime number.

So, why 91 is not a prime number is because it has more than two factors that contradict the definition of prime numbers.

91 is a Prime or Composite Number

As we can see why is 91 not a prime number it is because it has more than two factors other than 1 and the number itself. It contradicts the definition of prime numbers. So 91 is a composite number because composite numbers have factors more than 2.

When we factorize 91 we get

91 ÷ 1 = 91

91 ÷ 7 = 13

91 ÷ 13 = 7

91 ÷ 91 = 1

Here we get four factors for 91 and they are 1, 13, 7, 91. As the total number of factors is more than two it satisfies the condition of a composite number. Hence 91 is a composite number.

What Kind of Number is 91 Then?

Now it is clear that 91 does not satisfy the conditions of prime numbers. It satisfies the conditions for composite numbers so 91 is a composite number.

Some other kinds of numbers in which 91 falls are as follows:

• A natural number

• A positive integer

• An odd number

• A rational number

• A composite number

• A whole number

Solved Examples

Example 1: Is 19 a Prime Number or not?

Solution:

We can check the number is prime or not in two ways.

Method 1:

The formula for the prime number is 6n + 1

Let us write the given number in the form of 6n + 1.

6(3) + 1 = 18 + 1 = 19

Method 2: 

Check for the factors of 19

19 has only two factors 1 and 19. 

Therefore, by both methods, we get 19 as a prime number.

Example 2: Is 53 is a prime number or not?

Solution:

Method 1:

To know the prime numbers greater than 40, the below formula can be used.

n₂ + n + 41, where n = 0, 1, 2, ….., 39

Put n= 3

32 + 3 + 41 = 9 + 3 + 41 = 53

Method 2:

53 has only factors 1 and 53.

So, 53 is a prime number by both the methods.

A polynomial is an algebraic expression that can have one or more terms. The meaning of “Poly” is “many” and “nominal” means “terms,” so, in other words, a polynomial is “many terms.” It can have constants, variables, and exponents. They all can be combined using mathematical operations like additions, subtraction, multiplication, and division except that a division by a variable is not allowed in a polynomial expression.

Example of a polynomial – 2xy² + 4x – 6 –> This polynomial has 3 terms which are 2xy², 4x, and 6

A polynomial can have one or more terms, but infinite numbers of terms are not allowed as well as the exponent can only be positive integers (0, 1, 2, …); hence 3xy-2 is not a polynomial.

Interpolation

Within a range of a discrete set of data points, interpolation is the method of finding new data points. It is a technique in which an estimate of a mathematical expression is found, taking any intermediate value for the independent variable. The main use of interpolation is to figure out what other data can exist outside of their collected data. Many professionals like photographers, scientists, mathematicians, or engineers use this method for their experiments. A common use is in the scaling of images when one interpolates the next position of a pixel based on the given positions of pixels in an image.

Lagrange Interpolation Theorem

This theorem is a means to construct a polynomial that goes through a desired set of points and takes certain values at arbitrary points. If a function f(x) is known at discrete points x_i, i = 0, 1, 2,… then this theorem gives the approximation formula for nth degree polynomials to the function f(x). More so, it gives a constructive proof of the theorem below:

For a point p(2,4), how do we represent it as a polynomial?

P(x) = 3

P(1) = 3

Similarly for a sequence of points, (2,3), (4,5) how can we find a polynomial to represent it?

P(x) = (x-4)/(2-4) * 3 + (x-2)/(4-2) * 5

P(2) = 3 and P(4) = 5

Going by the above examples, the general form of Lagrange Interpolation theorem can be gives as:

P(x) = (x – x₂) (x-x₃)/(x₁ – x₂) (x₁ – x₃) * y₁ + (x – x₁) (x-x₃)/(x₂ – x₁) (x₂ – x₃) * y₂ + (x – x₁) (x-x₂)/(x₃ – x₁) (x₃ – x₂) * y₃

Or P (x)= [sum_{i=1}^{3}] [P_{i}](x) [Y_{i}]                    

The Theorem – For n real values which are distinct: x₁, x₂, x₃, x₄,..x_n, and n real values (might not be distinct) y₁, y₂, y₃, y₄… y_n, a unique polynomial exists with real coefficients which satisfies the formula:

P(x_i) = y_i, i ϵ (1, 2,3, …, n) so that deg(P) < n

Proof of Lagrange Theorem

Let us consider an nth degree polynomial which is given by the below expression:

F(x) = A₀ (x-x₁) (x-x₂) (x-x₃)….(x-x_n) + A₁ (x-x₀) (x-x₂) (x-x₃)….(x-x_n) + ……..+ A_n (x-x₁) (x-x₂) (x-x₃)….(x-x_n-1)

Now we substitute values of our observations i.e. xi and we obtain the values of Ai:

So we put x = x₀ and we get A0 as below:

f(x₀) = y₀ = A₀ (x₀-x₁) (x₀-x₂) (x₀-x₂)….(x₀-x_n), the other terms become 0

Hence A₀ = y₀/(x₀-x₁) (x₀-x₂) (x₀-x₃)….(x₀-x_n)

Similarly for x1 we would get 

f(x₁) = y₁ =  = A₁ (x₁-x₀) (x₁-x₂) (x₁-x₃)….(x₁-x_n), the other terms become 0

Hence A₁ = y₁/(x₁-x₀) (x₁-x₂) (x₁-x₃)….(x₁-x_n)

In this way we can obtain all the values of As from A₂, A₃,… A_n

A_n = y_n/(x_n-x₀) (x_n-x₂) (x_n-x₃)….(x_n-x_n-1)

Now if we substitute all the values of As in the main function, we get Lagrange’s interpolation theorem:

F(x) =  y₀ * (x-x₁) (x-x₂) (x-x₃)….(x-x_n)/ (x₀-x₁) (x₀-x₂) (x₀-x3)….(x₀-x_n) + y₁ * (x-x₀) (x-x₂) (x-x₃)….(x-x_n)/ (x₁-x₀) (x₁-x₂) (x₁-x₃)….(x₁-x_n) + ……..+ y_n (x-x₁) (x-x₂) (x-x₃)….(x-x_n-1)/ (x_n–x₀) (x_n-x₂) (x_n-x₃)….(x_n-x_n-1)

Note: Lagrange’s theorem applies to both equally and non-equally spaced points. This means that all the values of x_s are not spaced equally.

Uses of Lagrange Interpolation Theorem – In science, a complicated function needs a lot of time and energy to be solved. This makes experiments difficult to run. In order to create a slightly less complex version of the original function, the interpolation method comes in use.

Conclusion 

The Lagrange theorem generalizes the well-established mathematical facts like a line is uniquely determined by 2 points, 3 points uniquely determine the graph of a quadratic polynomial, and so on. There is a caveat here i.e.; the points must have different x coordinates. The image enlargement technique uses principles of the Lagrange theorem in trying to describe the tendency of image data by using interpolation polynomials to estimate unknown data. This helps in image enlargement.

The theorem can be expressed in a mathematical formula as shown here: 

[P (x) = sum i = 1_{n}sum i = 1_{n}P_{i}P_{i}(x)]

Yi and it applies to all values of x, whether they are equally spaced or not.

Length area and volume, Dimensional measures of one-dimensional, two- dimensional, and three-dimensional geometric objects. All three are magnitudes, that represent the “size” of an object. We can define length as the size of a line segment (see distance formulas), the area is the size of a closed region in a plane, and volume is the size of a solid. 

Formulas for the area as well as the volume are based on lengths. For example, the area of a circle equals π times the square of the length of its radius (denoted by r), and the volume of a rectangular box is the product of its three linear dimensions that is: length, width, as well as height.

In this article, we are going to discuss length area and volume, volume with fractional edge lengths and unit cubes, differential length area and volume as well as measuring volume as area times length.

What is Length?

Out of all the three-length area and volume. Length is a measure of distance. In the International System of Quantities, we can define length as a quantity with dimension distance. In most systems of measurement, a base unit for length is chosen, and from this base unit, all other various units are derived. In the International System of Units that is the SI system, the base unit for length is the metre (m).

Length is known to mean the most extended dimension of any given fixed object. However, this is not always the case, and the length of an object may depend on the position the object is in.

Various terms for the length of any given fixed object are used, and these include height, we can define height as the vertical length or vertical extent, and width, breadth, or depth. The term height is used when there is a base from which vertical measurements can be taken. The width or breadth of any object usually refers to a shorter dimension when the length is the longest one. Whereas depth is used for the third dimension of a three-dimensional object.

Let’s solve a problem!

Question 1. Bridge A is 50 m long, Bridge B is 24 m long. Find the total length of both the bridges.

Solution. Length of Bridge A = 50 m, Length of Bridge B = 24 m 

Total length of bridge A and B  = (50m + 24m) = 74 m 

What is Area?

Out of all the three-length area and volume. Area can be defined as the region bounded by the shape of an object. The space covered by any figure or any geometric shape is known to be the area of the shape. The area of all the shapes depends upon their dimensions as well as their properties. Different shapes have different areas. For example, the area of the square is different from the area of the kite.

If two objects are known to have a similar shape then it’s not necessary that the area covered by them will be equal unless and until the dimensions of both shapes are also equal.  Let’s suppose, there are two rectangle boxes, which have the length as L1 and L2 as well as breadth equal to B1 and B2. So the areas of both the rectangular box say, Area1 and Area2 will be equal only if L1 equals L2 and B1 equals B2.

 

Most Common Area Formulas 

 

What is Volume?

Out of all three – length area volume. We can define volume as a quantity that specifies the space occupied by a three-dimensional shape or object. The volume of a cube can be defined as the cube of its edge length (side3). For example, if the edge length of any given cube is equal to 5 cm, then its volume will be:

V = 5*5*5 cm equals 125 cubic cm

 

Unit of Volume

Out of length area and volume, the Volume of a solid is generally measured in cubic units. For example, if the dimensions are given in meters, then the volume of any object will be in cubic meters. Cubic meters are known to be the standard unit of volume in the International System of Units (that is SI). Similarly, various units of volume are cubic centimetres, cubic inches, cubic foot, etc.

 

Volume of Shapes

Linear Equations in One Variable Definition: they are known to be those equations which are of the first order. These equations are written for the lines in our coordinate system.

Linear equations in one variable are also the first-degree of equations because it has the highest exponent of variables known as 1.

Some of the examples for such kind of equations are given below:

• 2x – 4 = 0, 

• 4y = 8

• m + 2 = 0,

• x/2 = 4

• x + y = 4

• 3x – y + z = 9

When the equation has a homogeneous variable (i.e. only one variable), therefore such type of equation is known as the Linear equations in one variable. In other words, a linear equation can be retained by relating zero to a linear polynomial over any of the field, from where the coefficients are obtained.

The solutions for the linear equations in one variable will create values, which when substituted for the unknown values, will make the equation true. In the case of 1 variable, there’s just one solution, like x + 2 = 0. But just in the case of the two-variable equation, the solutions are calculated because of the Cartesian coordinates of some extent of the Euclidean plane.

Solving Linear Equations

Solving equations works in much an equivalent way, but now we’ve to work out what goes into the x, rather than what goes into the box. Since we are older now than the time when we were filling in the boxes, the equations also can be much more complicated, and so the methods we’ll use to solve the given equations will be a little bit more advanced than expected.

Generally, to solve a linear equation for a given variable, we have to “undo” everything that has been done to the variable. We have to do this to get the variable by one’s own selves; when put in technical terms, we are “isolating” the known variable. This leads to the equation being rearranged to mention “(variable) equals (some number)”, where (some number) is the answer they’re trying to find . For instance:

The variable is the letter x. To solve this equation, we need to get the x by its own selves ; that is, we need to get x on one side of the “equals” sign, and some other number on the other side.

Since we would like just x on the one side, this suggests that we do not just like the “plus six” that’s currently on an equivalent side because the x. Since the 6 is added to the x, we have to subtract this 6 to obviate it. That is, we will be able to subtract a 6 from the x so as to “undo” their having added a 6 thereto.

Here’s the solution,

x + 6 = -3

x = -3 – 6

x = -9

Steps – by – Step Solution for a Linear Equations in One Variable

1. Simplify both sides of the equation.

2. Use addition or subtraction properties of equality to gather the variable terms on one side of the equation and therefore the constant terms on the opposite.

3. Use the multiplication or division properties of equality to form the coefficient of the variable term adequate to 1.

4. Check your answer by substituting your solution into the first equation.

Note: If when solving an equation, the variables are eliminated to reveal a real statement like 13 13, then the answer is all real numbers. This type of equation is called an identity. On the opposite hand, if the variables are eliminated to reveal a falsehood like 7 3, then there’s no solution. This type of equation is called a contradiction. All other linear equations in one variable which have just one solution are called conditional.

Solving Equations By Collecting Terms

Suppose we wish to unravel the equation 3x + 15 = x + 25 The important thing to recollect about any equation is that the sign represents a balance. What the sign says is that what’s on the left-hand side is strictly an equivalent as what’s on the right-hand side. So, if we do anything to at least one side of the equation we’ve to try to do it to the opposite side. If we don’t, the balance is disturbed.

Therefore, whatever operation we perform on either side of the equation, goodbye as it’s wiped out precisely the same way on all sides the balance is going to be preserved.

Our initiative in solving an equation is to aim to collect all the x’s together and to collect all the numbers together. From 3x + 15 = x + 25 we will subtract x from all sides, because this may remove it entirely from the proper, to offer 2x + 15 = 25 we will subtract 15 from all sides to offer 2x = 10 and eventually, by dividing all sides by 2 we obtain x = 5.

Therefore the solution for the equation is x = 5. This solution should be checked by substitution into the first equation so as to see that each side is an equivalent. If we do that, the left is 3(5) + 15 = 30. The right is 5 + 25 = 30. So the left equals the proper and that we have checked that the answer is correct.