# Equations

Top
 Sub Topics An equation is defined as a mathematical statement. An equation has two expressions, each expression is separated by an equal sign (=). An equation can have variables on both sides of the "=" sign.Given below are some examples on equations:$x = 52$$x = 5 + 2$$5x + 43 = 7x$$7x + 233 = 77 5x - 23 = 67 ## Solving Equations Back to Top Equations can be solved by following the steps given below: • Group the unknown variables to the left hand side of the equation by adding or subtracting terms on both the sides of the equation • Group the numerical terms to the right hand side of the equation • Divide both sides of the equation with the coefficient of the unknown variable. • The result is the value of the unknown variable. ### Examples on Solving Equations: Given below are some examples that explain how to solve equations. Example 1: 7x = 6x +1 Solution: 7x = 6x +1 7x - 6x = 6x - 6x + 1 (Subtract 6x from both the sides of the equation.) x = 1 Example 2: 12x - 6 = 6x + 12 Solution: 12x - 6 = 6x + 12 12x - 6x - 6 = 6x -6x + 12 (Subtract 6x from both the sides of the equation.) 6x - 6 = 12 6x - 6 + 6 = 12 + 6 (Subtract 6 from both sides of the equation) 6x = 18 \frac{6x}{6} = \frac{18}{6} (Divide by 6 on both sides) x = 3 Example 3: 3x + 5x = 4 Solution: 3x + 5x = 4 8x = 4 x = \frac{1}{2} ## System of Equations Back to Top System of Equations can be defined as a set of two or more equations with the common variables graphed on the same coordinate plane. They are also known as simultaneous equations. Different methods can be adopted to find the value of the variables and hence solve the system of equations. There can be mainly three type of solutions: • Exactly One Solution • Infinite Solution • No Solution Exactly one solution If the slopes of the equations are different, then they will have only one solution. For example: x + y = 10 2x + y = 5 Here, x + y = 10 can be written as y = -x + 10 and slope = -1 Here, 2x + y = 5 can be written as y = -2x + 5 and slope = -2 As slopes are different, they will have only exactly one solution Infinite solution If the slopes and y-intercepts are same, then they will have no solution. For example: x + 2y = 2 2x + 4y = 4 Here, x + 2y = 2 can be written as y = \frac{-x}{2} + 1 and slope = \frac{-1}{2} and y intercept =1 Here, 2x + 4y = 4 can be written as y = \frac{-x}{2} + 1 and slope = \frac{-1}{2} and y intercept =1 No solution If the slopes are the same but y-intercepts are different, then they will have infinite solutions. For example: y - x = 1 2y - 2x = 4 Here, y - x = 1 can be written as y = x + 1 and slope = 1 and y intercept =1 Here, 2y - 2x = 4 can be written as y = x + 2 and slope = 1 and y intercept =2 Here, the slopes are the same but the intercepts are different. So, it has infinite solutions. The most commonly used methods are as follows: • Substitution Method • Elimination Method • Graphical Method Substitution Method Here, an equation is solved for any one variable and then it is substituted in the other equation for solving the rest of the variable. ### Example on Substitution Method -x + 3y = 11……….. (1) 4 x - y = -11………… (2) Solve for x in equation (1) -x + 3y =11 => -x = 11 - 3y => x = 3y - 11 Plug this in equation (2) 4(3y - 11) - y = -11 12y - 44 - y = -11 11y - 44 = -11 11y = -11 + 44 11y = 33 y = \frac{33}{11} =3 So, x = 3(3) - 11 = 9 - 11 = -2 Hence, the solution is (-2, 3) Elimination Method This method is also called as the addition method. Here, the main concept is to replace the coefficients of an equation with a combination of the coefficients of the equation in the system. For that each equation must be multiplied with a constant and added or subtracted with the other equation, so that any of the variables may be eliminated. By this, the equation can be solved. ### Example on Elimination Method -x + 3y = 11……….. (1) 4 x - y = -11………… (2) Multiply 4 by (1) -4x + 12y = 44………. (3) Add (2) + (3) 4x - y = -11 -4x + 12y = 44 0x + 11y = 33 So, y = \frac{33}{11} =3 Hence (1) => -x + 3y = 11 => -x + 3 (3) =11=> -x + 9 = 11 => -x = 11 - 9 => -x = 2 => x = -2 So, the solution is (-2, 3) Algebraic Equation Method Here, the equation can be equated by setting it up equal to one another, and hence solve for the variables. ### Example on Algebraic Equation Method -x + 3y = \frac{55}{4}……….. (1) 4x - y = -11………… (2) Solve for x in (1) => -x + 3y = \frac{55}{4} -x = \frac{55}{4} - 3y x = 3y - \frac{55}{4} Solve for x in (2) =>4x - y = -11 4x = -11+ y x = \frac{-11}{4} + \frac{y}{4} Equating both 3y - \frac{55}{4} = \frac{-11}{4} + \frac{y}{4} Add 11 on both sides 3y = \frac{-11}{4} + \frac{55}{4} + \frac{y}{4} 3y = \frac{44}{4} + \frac{y}{4} Subtract \frac{y}{4} on both sides 3y - \frac{y}{4} = \frac{44}{4} \frac{11y}{4} = \frac{44}{4} Multiply 4 on both sides 11y = 44 Divide by 11 y = \frac{44}{11} = 4 Substitute it in (1) (-x) + 3y = \frac{55}{4} (-x) + 3 * (4) = \frac{55}{4} (-x) +12 = \frac{55}{4} (-x) = \frac{55}{4} - 12 (-x) = \frac{7}{4} x = \frac{-7}{4} So, the solution is (\frac{-7}{4} , 4) Graphical Method Here, each equation can be graphed by plotting the corresponding points of the graph. The points where they meet form the solution of the equation and using that it will be possible to find the nature of the solution. ### Example on Graphical Method -x + 3y = 11……….. (1) 4x - y = -11………… (2) For the equations: -x + 3y = 11 Add x on both sides -x + 3y + x = 11 + x 3y = 11 + x Divide with 3 on both sides \frac{3y}{3} = \frac{11}{3} + \frac{x}{3} y = \frac{11}{3} + \frac{x}{3} Now let us assign some values and find the corresponding y-values to plot on the graph  x 0 -2 4 y = \frac{11}{3} + \frac{x}{3} y = \frac{11}{3} + \frac{x}{3}y = \frac{11}{3} + \frac{0}{3}y = \frac{11}{3}y = 3.667 y = \frac{11}{3} + \frac{x}{3}y = \frac{11}{3} + \frac{-2}{3}y = 3.667 - 0.667y = 3 y = \frac{11}{3} + \frac{x}{3}y = \frac{11}{3} + \frac{4}{3}y = 3.667 + 1.333y = 5 For the equation: 4x - y = -11 Subtract 4x on both sides 4x - y - 4x = -11 - 4x -y = -11 - 4x Multiply with (-1) on both sides (-1)\times(-y) = (-1)\times(-11 - 4x) y = 11 + 4x y = 4x + 11  x 0 -2 4 y = 4x + 11 y = 4x + 11 = 4(0) + 11= 11 y = 4x + 11 = 4(-2) + 11= 3 y = 4x + 11 = 4(4) + 11 =27 Now let us plot these ordered pairs We can see that, it intersects at (-2, 3). So, (-2,3) is the solution. ## How to Write an Equation? Back to Top There are a few steps to be followed to write an equation. They are as follows: • Find what is to be found and assign a variable to it. • Check what is the other data provided. Adequate data is needed for solving a variable. • Establish a relation between the given data and the variable. It may be done by translating the words provided into mathematical symbols. • Now form the equation such that the variable is equal to the combination of the provided data. ### Examples on writing an equation Given below are some examples on writing an equation. Example 1: The quotient of five hundred and fifty more than a number is hundred. Solution: First find what is to be found and assign a variable to it. Here, we need to find a number and assign it as “x”. The other data provided is five hundred, fifty and hundred. Establish a relation between the given data and the variable. It may be done by translating the words provided into mathematical symbols. First, find quotient of five hundred and fifty more than a number = \frac{500}{(50+x)} This is equated to 100. So, \frac{500}{(50+x)} =100 Solve for the variable. 500 = 100 (50 + x) 500 = 5000 + 100x So, 100x = 500 - 5000 100x = -4500 x = \frac{-4500}{100} x = -45 Write the following sentences as an algebraic equation. • A number increased by three is ten: x+3 =10 • Mary is x years old. In twenty years she will be thirty one years old: x + 20 = 31 • Thirteen is forty one less than six times a number: 13 = 6x - 41 ## Two Step Equations Back to Top Two Step Equations are defined as solving an equation two times. This is called so because, there are only two operations needed to solve the equation. Facts to remember 1. Opposite of addition operation is subtraction and vice versa. 2. Opposite of multiplication operation is division and vice versa. Two types of such equations can be defined • 2-Step Multiplication Equations • 2-Step Division Equations ### 2-Step Multiplication Equations The process for solving 2 step multiplication equations is as follows: • Add or subtract terms from both sides, so that, the Coefficient and Variables are on either side of the equal sign. • Divide both sides by the coefficient of the variable. • Solve the equation. ### Examples on 2-Step Multiplication Equations Given below are some examples on 2-step multiplication equations. Example 1: 8x + 5 = 21 Solution: To solve, the variable must be on one side and all the other numbers must be on the other side: 8x + 5 = 21 Subtract 5 from both the sides 8x + 5 = 21 - 5 - 5 8x + 0 = 16 8x = 16 Divide both the sides by 8 \frac{8x}{8} = \frac{16}{8} x = 2 Example 2: 3y – 10 = 44 Solution: Add 10 to both the sides 3y – 10 + 10 = 44 + 10 3y = 54 Divide both the sides by 3 y = 18 ### 2-Step Division Equations The process for solving a 2 step division equation is as follows: • Add or subtract terms from both sides, so that, the Coefficient and Variables are on either side of the equal sign. • Multiply both the sides by the coefficient of the variable. • Solve the equation. ### Examples on 2-Step Division Equations Given below are some examples on 2-step division equations. Example 1: \frac{x}{3} - 5 = 22 Solution: To solve, the variable must be on one side and all the other numbers on the other side. \frac{x}{3} - 5 = 22 Add 5 on both the sides \frac{x}{3} - 5 + 5 = 22 + 5 \frac{x}{3} = 27 Multiply both the sides by 3 \frac{3x}{3} = 3 × 27 x = 81 Example 2: \frac{y}{4} + 7 = 14 Solution: Subtract 7 from both the sides \frac{y}{4} + 7 - 7 = 14 - 7 \frac{y}{4} = 7 Multiply both the sides by 4 y = 28 ## Multi Step Equations Back to Top A multi-step equation can be defined as an equation that requires more than two operations as steps to solve it. In order to solve a multi step equation, the equation can be simplified by using the distributive property and combining like terms depending on the requirement. ### Examples on Solving Multi Step Equations Given below are some examples on solving multi step equations. Example 1: n - 3 = 5 + 2n Solution: The basic idea is to get the variables to the left and the numbers to the right. For that subtract 2n from both sides n – 3 = 5 + 2n - 2n - 2n -n – 3 = 5 Add 3 on both sides -n – 3 = 5 + 3 +3 -n = 8 Multiply (-1) on both sides (-1)(-n) = (-1) 8 n = - 8 Example 2: Find three consecutive odd integers whose sum is 99 Solution: To solve, lets assume the smallest odd integer is x. Then, the next two consecutive odd integers are x+2 and x + 4. Sum of all integers is 99. So, x+ (x+2) + (x + 4) = 99 Open parenthesis x + x + 2 + x + 4 = 99 Combine like terms 3x + 6 = 99 Subtract 6 from both sides 3x + 6 = 99 - 6 -6 3x = 93 Divide both sides by 3 \frac{3x}{3} = \frac{93}{3} x = 31 x + 2 = 33 x + 4 = 35 So, the three consecutive odd integers are 31, 33 and 35 In general, the steps followed in solving multi step equations are as follows: • Isolate the variable and hence the solution of equation can be obtained • Use addition as inverse of subtraction and vice versa. • Use division as inverse of multiplication and vice versa. The numbers / constants used can be integers, fractions or decimals. Lets see example for each ### With integer: 2(x + 1) – (x + 3) = 5 First use the distribution property and open the parenthesis 2x + 2 – x – 3 = 5 Combine like term 2x – x + 2 – 3 = 5 x – 1 = 5 Add 1 on both the sides x – 1 = 5 + 1 = +1 x = 6 ### With fractions: To solve with fraction, first of all simplify the equation without fraction (\frac{x}{4} + 3) - 3 =6 Add 3 on both the sides (\frac{x}{4} + 3) - 3 = 6 +3 +3 \frac{x}{4} + 3 = 9 Subtract 3 from both the sides \frac{x}{4} + 3 =9 - 3 -3 \frac{x}{4} = 6 Multiply 4 on both the sides \frac{x}{4} (4) = 6(4) x = 24 ### With decimals: Simplify as much as possible 3 (x - 0.5) - 4.5 = 3.5 + x Open parenthesis 3x – 1.5 – 4.5 = 3.5 + x 3x – 6 = 3.5 + x Subtract x from both the sides 3x – 6 = 3.5 + x - x = - x 2x - 6 = 3.5 Add 6 to both the sides 2x - 6 = 3.5 + 6 + 6 2x = 9.5 Divide both the sides by 2 \frac{2x}{2} = \frac{9.5}{2} x = 4.75 ## Graphing Equations Back to Top Graphing an equation is more of a visual way of looking at an equation so that its possible to understand its structure and its properties. It is drawn with axes perpendicular to each other. In graphing a linear equation, two axes, x (horizontal axis) and y (vertical axis), perpendicular to each other, are taken. They intersect a point called as the origin. It is called coordinate plane or rectangular coordinate plane. Each point on the plane is represented by an ordered pair (a, b) where a is a point on the x axis and b is a point on the y axis. There will be infinite points for a linear equation. But, its not possible to determine all of them. So we use minimum points to get the whole equation graphed correctly. A linear equation can be mostly drawn in two ways. • Using slope and y intercept • Using T chart ### Using slope and y intercept The standard form of writing any linear equation is y = (m) * x + b, where, m, being the x coefficient, is called the slope and b, being the constant, is called the y intercept. The following procedure has to be followed: • Write the given equation in the slope intercept form. • Recognize the y intercept and plot it on the graph. • Recognize the slope and using it determine another point on the xy plane. • Join the two points and also extend it both to the either sides of the points to get the required line ### Examples on Graphing Equations Here are some examples on graphing equations Example 1: Graph 3y - 4x = 12 Solution: First write it in the slope intercept form 3y - 4x = 12 Add 4x on both the sides So 3y = 4x +12 Divide both the sides by 3 y = (\frac{4}{3}) x + 4 So slope m = \frac{4}{3} and y intercept = (0, 4) Plot the point (0, 4) on the graph. From 4 go 4 units up, so it goes to 8 and 3 units right, so we end up at (8, 3) Hence, draw a line joining the two points. Example 2: Graph 2y + 3x = 8 Solution: Write in slope intercept form y = (\frac{-3}{2})x + 4 Slope = \frac{-3}{2} y intercept = (0, 4) Plot the point (0, 4) on the graph. From 4 go -3 units down, so it goes to 1 and 3 units right, so we end up at (2, 1) ### Using T chart When graphing any linear equation, first form a chart, known as the T- chart. It is called so because of the shape it takes. It will have two column=> left column contains x points and right column contains y points. You can choose any reasonable x values, plug it in the equation and solve for y. Grouping the x and corresponding y values makes a pair that can be used for plotting. Points to remember. • Any x values can be chosen such that the y value exists. • For any straight line we need at least three points. Draw a line joining all the three points. Example 3: Graph y = 5x + 1 Solution: For x - values assume -1, 0, 1 Lets form the T chart  x y = 5x + 1 ( x, y ) -1 y = 5(-1) + 1 = -4 ( 1, -4 ) 0 y = 5(0) + 1 = 1 ( 0, 1 ) 1 y = 5(1) + 1 = 6 ( 1, 6 ) Plot this point on the graph and drawn the line joining the points. Example 4: Graph y = 3x + 4 Solution: Lets form the T chart  x y = 3x + 4 ( x, y ) 0 y = 3(0) + 4 = 0 + 4 = 4 ( 0, 4 ) 1 y = 3(1) + 4 = 3 + 4 = 7 ( 1, 7 ) 2 y = 3(2) + 4 = 6 + 4 = 10 ( 2, 10 ) Plot this point on the graph and draw a line joining the points. ## Math Inequality Back to Top Any expression that contains symbols like <, >, \leq and \geq is called an inequality. • < : less than • > : greater than • \geq : greater than or equal to • \leq : less than or equal to ### Examples on Inequalities Given below are some of the examples on inequalities: • 3x + 12 > 9 • x < 7 • x > -10 Example 1: Solve 3x + 5 < 11 Solution: The first priority is to isolate x. For that, subtract 5 from both sides 3x + 5 - 5 < 11 - 5 3x < 6 Divide both sides by 3 \frac{3x}{3} < \frac{6}{3} So, the solution is x < 2 ## Simplifying Equations Back to Top Simplifying an equation is process of solving an equation. Any equation can be solved easily if the equation is simplified. In simplifying equations, operations can be performed in different ways as they are said to possess different properties: Addition is commutative: x + y = y + x Addition is associative: x + ( y + z ) = ( x + y ) + z Multiplication is commutative: x × y = y × x Multiplication is associative: (x × y) × z = x × (y × z) There are no standard steps that can be followed to simplify an equation but there are certain methods that can be used. These methods can only be adopted according to the requirement of each question. To understand when to adopt these methods, requires practice. Let us examine some of these methods: ### Cross Multiplication This is a process by which the denominator terms can be taken to the numerator in an equation. The basic idea is if \frac{a}{b} = \frac{c}{d} then, ab = cb ### Examples on Cross Multiplication Given below are some examples on cross multiplication. Example 1: \frac{3}{9} = \frac{12}{x} Solution: Here, the main purpose is to solve for x. But, here x is in the denominator. To simplify, the best way is cross multiplication. So, \frac{3}{9} = \frac{12}{x} 3x = 12 × 9 3x = 108 x = \frac{108}{3} x = 36 Example 2: Solve \frac{2}{3} = \frac{u}{6} Solution: Here, first we can cross multiply 2 × 6 = u × 3 12 = 3u Divide both sides by 3 4 = u u = 4 ### Differentiate between like and unlike terms Like terms are those that have the same variables, whereas unlike terms have different variables. Like terms can be combined together and so simplified whereas unlike terms cannot be combined. Simplification is very simple if we combine like terms. For example: x + 2x = 3x 3a - 4a = -a 6xy - 6xy = 0 ### Examples on Simplifying Equations Given below are some examples on simplifying equations. Example 1: Simplify 6a + 4xy - 7yx + 5a = 0 Solution: Here, there are four terms of which 6a and 5a are like terms. So, they can be combined. Now, xy = yx (Since, multiplication is commutative.) And so, 4xy and 7xy becomes like terms. So, 7yx = 7xy 6a + 4xy - 7yx + 5a = 0 6a + 4xy - 7xy + 5a = 0 (6a + 5a) + (4xy - 7xy) =0 (11a) + (-3xy) = 0 11a - 3xy = 0 Another important rule that the operation has to follow is called in short as PEMDAS. It is a short form for order of operations. P – Parenthesis E – Exponents MD – multiplication or division AS – addition or subtraction According to the rule, in any equation first do the parenthesis then the exponent, if any. Operations has to performed from left to right with multiplication or division, which ever comes first and addition or subtraction, which ever comes first. Example 2: Simplify 7x + (6x × 52 + 3x)=0 Solution: Here, start with parenthesis (6x × 52 + 3x) In that, take the exponent first 52 = 25 So (6x × 52 + 3x) = (6x × 25 + 3x) (6x × 52 + 3x) = (6x × 25 + 3x) = (150x + 3x) = 153x 7x + (6x × 52 + 3x)=0 7x + 153x = 0 160x = 0 ## Linear Equations Back to Top Many systems in all kinds of diverse fields can be characterized by the input-output analysis. Generally, to analyse such systems, we follow three different methods: Method 1: The system is described by some specific mathematical form, in which the input is given and an output is to be found. Method 2: The system and the output are provided and we need to find the input. Method 3: Both the input and the output are provided and we need to create the system. This system of finding the solution from the given input and output gives us the linear system or equations. The general method for solving a linear equation is to perform algebraic operations on the system that will not alter the solution set and will produce a successive simpler system, until a point is reached where it can be checked whether the system is consistent or not and what the solutions are. A finite set of linear equations are called a system of linear equations or more accurately a linear system. The variables in these are unknowns. The algebraic operations will follow as mentioned: 1. Multiply an equation by a non-zero constant 2. Interchange the two equations 3. Add constant times of one equation to another A linear equation will not involve any products and or roots of variables and all these variables occur only to the first power and do not appear for the assessment of logarithmic, trigonometric, or even exponential functions. Few examples of linear equations are given below: 3x + 5y = 12 \frac{1}{2}$$x + 7y + 2z = -9$

$x_{1} - 2x_{2} - 3x_{3} + x_{4} = 0$
The following are not considered as linear equations:
$5x – 2y – xy = 7$

$Sin x + y = 0$

$2x + 5y^{2} = 6$

The quadratic equation given is $ax^2 + bx + c = 0$, where a, b, and c are real numbers with a ≠ 0, the solutions would be given by the quadratic formula $x$ = $\frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$. The expression within the radical sign, $b^2 – 4ac$ is called the discriminant and it will enable us to discriminate among the possibilities for the solution.
• If $b^2 – 4ac > 0$, the quadratic formula yields two real solutions.
• If $b^2 – 4ac = 0$, the quadratic formula yields single repeated solution
• If $b^2 – 4ac < 0$, the quadratic formula yields two complex numbers as solutions, which are nothing but a complex conjugate pair.