9,000 equations in 567 variables, 4. etc. An equation such as y=x+7 is linear and there are an infinite number of ordered pairs of x and y that satisfy the equation. Linear Equations in the Real World. After each click the graph will be redrawn and the … However, the word linear in linear equation means that all terms with variables are first degree. C (x) is a cost function. Solving one step equations. P (x) is a profit function. In this article, we are going to learn how to solve systems of linear equations using the commonly used methods , namely substitution and elimination. So a System of Equations could have many equations and many variables. Linear function interactive app (explanation below): Here we have an application that let's you change the slope and y-intercept for a line on the (x, y) plane. There are several systems of linear equations involving the same set of variables. There are several methods of solving systems of linear equations. All right reserved. 4x−7(2−x) =3x+2 4 x − 7 (2 − x) = 3 x + 2 Solution 2(w+3)−10 = 6(32−3w) 2 … 1. Solution: Let’s rewrite it as ordered pairs(two of them). Linear Equations: Solutions Using Elimination with Three Variables Systems of equations with three variables are only slightly more complicated to solve than those with two variables. Scroll down the page for more examples and solutions. A system of linear equations a 11 x 1 + a 12 x 2 + … + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + … + a 2 n x n = b 2 ⋯ a m 1 x 1 + a m 2 x 2 + … + a m n x n = b m can be represented as the matrix equation A ⋅ x → = b → , where A is the coefficient matrix, It is also known as the slope and gives the rate of change of the dependent variable. Section 2-2 : Linear Equations Solve each of the following equations and check your answer. 6 equations in 4 variables, 3. We will only use it to inform you about new math lessons. Linear equations can be a useful tool for comparing rates of pay. In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. The general solution of the differential equation is expressed as follows: y = ∫ u(x)f (x)dx+C u(x), where C is an arbitrary constant. Well, a set of linear equations with have two or more variables is known systems of equations. A system of linear equations a 11 x 1 + a 12 x 2 + … + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + … + a 2 n x n = b 2 ⋯ a m 1 x 1 + a m 2 x 2 + … + a m n x n = b m can be represented as the matrix equation A ⋅ x → = b → , where A is the coefficient matrix, 5x-6=3x-8. A system of linear equations, written in the matrix form as AX = B, is consistent if and only if the rank of the coefficient matrix is equal to the rank of the augmented matrix; that is, ρ ( A) = ρ ([ A | B]). To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations i… \frac{x}{3}+\frac{x}{2}=10. Examples. simple and easy to handle mathematically. 5b = -2b + 3. Well, a set of linear equations with have two or more variables is known systems of equations. By using this website, you agree to our Cookie Policy. Sum and product of the roots of a quadratic equations Algebraic identities \frac{3}{4}x+\frac{5}{6}=5x-\frac{125}{3} \sqrt{2}x-\sqrt{3}=\sqrt{5} 7y+5-3y+1=2y+2. Often, the terms linear equation and linear function are confused. Examples of Linear Equations The simplest linear equation is the one with one variable: ax + b = 0. Linear equations can always be manipulated to take this form: $$ ax+b=0 $$ A system here refers to when you have two or more equations working together. 2x-4=10. Geometrically, these subspaces are points, lines, planes and spaces that pass through the point 0. Solving Systems of Non-linear Equations. To find the unique solution to a system of linear equations, we must find a numerical value for each variable in the system that will satisfy all equations i… The only thing different is the function notation. Some examples of a linear equation are shown in the image below. The above linear equation is only true if x = 5 and hence the given linear equation has only one solution i.e. In linear equation, each term is either a … Using the table, we can verify the linear function, by examining the values of x and y. Examples No.1 x + 6 = 8 is a linear equation. Then you can be expected that the equations have one solution. Solving Linear Equations in Two Variables. A little bit of algebraic manipulation makes it clear that the unique solution to this linear equation is always -b/a. y = 2x + 5 with a = 2 and b = 5, y = -3x + 2 with a = -3 and b = 2, and y = 4x + - 1 with a = 4 and b = -1 are other examples of linear equations. = 2r. The independent variable is x and the dependent variable is y. The standard form is ax² + bx + c = 0 with a, b, and c being constants, or numerical coefficients, and x is an unknown variable. solving equations This sections illustrates the process of solving equations of various forms. How to solve a nonlinear system when one equation in the system is nonlinear. m = y 2 − y 1 x 2 − x 1. x 2 ≠ x 1. linear-equation-calculator. Sum and product of the roots of a quadratic equations Algebraic identities en. X-2Y +3Z=9-X+3Y-Z=-6. Multiplying the left side of the equation by the integrating factor u(x) converts the left side into the derivative of the product y(x)u(x). There can be any combination: 1. x = 5. Create printable worksheets for solving linear equations (pre-algebra or algebra 1), as PDF or html files. The independent variable is x and the dependent variable is y. a is the constant term or the y intercept. A linear equation can have 1, 2, 3, or more variables. It also shows you how to check your answer three different ways: algebraically, graphically, and using the concept of equivalence.The following table is a partial lists of typical equations. Solving quadratic equations by quadratic formula. Example 1: Consider the equation 7x – 35 = 0. So let's say I had the equation 5-- a big fat 5, 5x equals 20. Example III What is its profit if it sells (a) 75 items, (b)150 items, and (c) 200 items? Solving quadratic equations by quadratic formula. (The equation in example I was z = 0, and the equation in example II was x = y.) Real world linear equations in action as well as free worksheet that goes hand in hand with this page's real world ,word problems. View Lecture 1 math.pdf from MATH 105 at Arab Academy for Science, Technology & Maritime Transport. let C = total cost, C = fixed cost plus variable cost = 7,000 + 600 x. In y = ax + b, x is called independent variable and y is called dependent variable. Example III Linear equations are all equations that have the following form: y = ax + b. 4r − 3. . 3(x + 5) = 2(− 6 − x) − 2x. A “system of equations” is a collection of two or more equations that are solved simultaneously.Previously, I have gone over a few examples showing how to solve a system of linear equations using substitution and elimination methods. \frac{3}{4}x+\frac{5}{6}=5x-\frac{125}{3} \sqrt{2}x-\sqrt{3}=\sqrt{5} 7y+5-3y+1=2y+2. The above linear equation is only true if x = 5 and hence the given linear equation has only one solution i.e. Nature of the roots of a quadratic equations. So let's start doing some problems. Intro to slope. 2 equations in 3 variables, 2. There can be any combination: 1. We apply the theorem in the following examples. of $25 per item and a fixed cost of $1600. y = 25 + 5(3) = 40. C (x) = fixed cost + variable cost. Examples, solutions, videos, and lessons to help Grade 8 students learn how to interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; … Slope. It has a variable cost If you studied the writing equations unit, you learned how to write equations given two points and given slope and a point. Linear Functions. The first company's offer is expressed as 450 = 40x. A system of linear equationsconsists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. (The word linear in linear function means the graph is a line.) 6 equations in 4 variables, 3. (Opens a modal) Slope & direction of a line. In the given equation, the value of the variable which makes L.H.S = R.H.S is called the solution of linear equation. A linear equation in two variables has three entities as denoted in the following example: 10x - 3y = 5 and 2x + 4y = 7 are representative forms of linear equations in two variables. https://courses.lumenlearning.com/.../chapter/introduction-linear-functions It has many important applications. P(75) = 20(75) - 1600 = -100 a Solving quadratic equations by factoring. and R.H.S. Solving linear equations using cross multiplication method. Positive & negative … If you can solve these problems with no help, you must be a genius! a and b are called constants. The coefficient of (or , or , or any letter) is the number in … Linear functions have a constant slope, so nonlinear functions have a slope that varies between points. f(2) =-4 and f(5) = -3 (2, -4) (5, … variable when x = 0. b is the coefficient of the independent variable. 5 = 2x + 3. If you studied the writing equations unit, you learned how to write equations given two points and given slope and a point. 5 = 2x + 3. The two most straightforward methods of solving these types of equations … A differential equation of type \[y’ + a\left( x \right)y = f\left( x \right),\] where \(a\left( x \right)\) and \(f\left( x \right)\) are continuous functions of \(x,\) is called a linear nonhomogeneous differential equation of first order.We consider two methods of solving linear differential equations of first order: Non-homogeneous Linear Equations . 2 equations in 3 variables, 2. There are several methods of solving systems of linear equations. Varying terms are numbers like , , or , … What is Linear Equation?. Linear equations can be added together, multiplied or divided. Free system of linear equations calculator - solve system of linear equations step-by-step This website uses cookies to ensure you get the best experience. For example, if one company offers to pay you $450 per week and the other offers $10 per hour, and both ask you to work 40 hours per week, which company is offering the better rate of pay? Customize the worksheets to include one-step, two-step, or multi-step equations, variable on both sides, parenthesis, and more. Example 2: Consider the equation 9(x – 1) – 35 = 8x + 37. It is attractive because it is (The equation in example I was z = 0, and the equation in example II was x = y.) Linear function vs. About me :: Privacy policy :: Disclaimer :: Awards :: DonateFacebook page :: Pinterest pins, Copyright Â© 2008-2019. A function assigns exactly one output to each input of a … Graph Linear Equations by Plotting Points It takes only 2 points to draw a graph of a straight line. \frac {r-3} {4}=2r. While solving a linear equation in two variables, one must always abide by the following rules. x = 5. 2X + Y=6. A x + B y = C , {\displaystyle Ax+By=C,} We’ll start off the solving portion of this chapter by solving linear equations. Graph the linear equation x = 4. For example, \(y=6x+2\) is linear because it has no squares, cubes, square roots, sines, etc. The slope of a line passing through points (x1,y1) and (x2,y2) is given by. a x + b y + c = 0 , {\displaystyle ax+by+c=0,} where the variables are x and y, and the coefficients are a, b and c . The calculator easily performs equivalent operations on the given linear system. Find 2 points which satisfy the equation, 3. Geometrically, these subspaces are points, lines, planes and spaces that pass through the point 0. Examples Relating to Three Variable Linear Equations. See linear equations in our everyday lives. We are going to use this same skill when working with functions. Top-notch introduction to physics. It is possible, as we’ll see in an example, to have these values show up in the solution set. Slope formula. 5x-6=3x-8. In general, any subset of the real coordinate space R n that is defined by a system of homogeneous linear equations will yield a subspace. R (x) is a revenue function. View Lecture 1 math.pdf from MATH 105 at Arab Academy for Science, Technology & Maritime Transport. then In y = ax + b, x is called independent variable and y is called dependent variable. In this article, we are going to learn how to solve systems of linear equations using the commonly used methods , namely substitution and elimination. General Form. 3X - Y= 4. then It is not necessary to write equations in the basic form. So at first this might look a little unfamiliar for you, but if I were to rephrase this, I think you'll realize this is a pretty easy problem. On solving we have 7x = 35 or x = 5. Then you can be expected that the equations have one solution. Since a linear function must be both linear and a function, we do not have a linear function here. The number of equations and the number of unknowns should be equal, and the equation should be linear (and linear independent). 2x-4=10. The linear function is popular in economics. Both are polynomials. Check the equation for varying terms and constant terms. And there is also the General Form of the equation of a straight line: … Slope. So a System of Equations could have many equations and many variables. A “system of equations” is a collection of two or more equations that are solved simultaneously.Previously, I have gone over a few examples showing how to solve a system of linear equations using substitution and elimination methods. An equivalent equation (that is an equation with exactly the same solutions) is. A linear equation can help you figure it out! Linear Function Examples. For example, 3x - 4y + 5z = 3 is a linear equation because the variables x, y, z are linear, but xy + 3z = 7 is not linear because of the term xy, which is a product of two variables. See linear equations in our everyday lives. Example 1 Solve each of the following equations. The number of equations and the number of unknowns should be equal, and the equation should be linear (and linear independent). A linear function has the following form y = f (x) = a + bx A linear function has one independent variable and one dependent variable. A linear function has one independent variable and one dependent variable. Welcome to level one linear equations. Example 1.29 It is the value of the dependent Nature of the roots of a quadratic equations. Slope formula. In order to investigate situations such as that of the skateboard manufacturer, we need to recognize that we are dealing with more than one variable and likely more than one equation. Linear Equations 1 Definition The general form of a linear equation is: Ax + By = C Examples: P (x) = R (x) - C (x) x = the number of items produced and sold. A function is an equation that has only one answer for y for every x. One stop resource to a deep understanding of important concepts in physics, Area of irregular shapesMath problem solver. What is total cost at varying levels of output? Examples of Quadratic Equation A quadratic equation is an equation of the second degree, meaning it contains at least one term that is squared. \frac{x}{3}+\frac{x}{2}=10. Graphing of linear functions needs to learn linear equations in two variables. Create printable worksheets for solving linear equations (pre-algebra or algebra 1), as PDF or html files. A linear equation in two variables has three entities as denoted in the following example: 10x - 3y = 5 and 2x + 4y = 7 are representative forms of linear equations in two variables. let x = units of output Solving quadratic equations by factoring. Examples, solutions, videos, and lessons to help Grade 8 students learn how to interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; … Solving quadratic equations by completing square. Section 2-2 : Linear Equations. A function notation ordered pair. Examples. For the linear function, the rate of change of y with respect the variable x remains constant. A simple example of addition of linear equations, R(x) = selling price (number of items sold), x = the number of items produced and sold. (a,b) = (2,5) f (a) = y coordinate, a=2 and y = 5, f (2) = 5. Linear Equations in the Real World. More precisely, a linear equation is one that is dependent only on constants and a variable raised to the first power. Basic-mathematics.com. Divide both sides by the coefficient of . Examples. Solving Systems of Non-linear Equations. You change these values by clicking on the '+' and '-' buttons. (Opens a modal) Slope & direction of a line. In the case of two variables, any linear equation can be put in the form. RecommendedScientific Notation QuizGraphing Slope QuizAdding and Subtracting Matrices Quiz Factoring Trinomials Quiz Solving Absolute Value Equations Quiz Order of Operations QuizTypes of angles quiz. u(x) = exp(∫ a(x)dx). Intro to slope. Everything you need to prepare for an important exam!K-12 tests, GED math test, basic math tests, geometry tests, algebra tests. X+2Y+3Z=-7. The slope, m, is here 1 and our b (y-intercept) is 7. In linear equation, the sign of equality (=) divides the equation into two sides such as L.H.S. Everything you need to prepare for an important exam! Linear Equations 1 Definition The general form of a linear equation is: Ax + By = C Examples: A simple example of addition of linear equations. A system of linear equationsconsists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. Real Life Math SkillsLearn about investing money, budgeting your money, paying taxes, mortgage loans, and even the math involved in playing baseball. It is not necessary to write equations in the basic form. In our example above, x is the independent variable and y is the dependent variable. Connect the points with a straight line, let x = 1 This is … Let’s take a look at some examples. It is considered a linear system because all the equations in the set are lines. In this example, the top equation is linear. Linear Equations With one Solution Example 1: Consider the equation 7 x – 35 = 0. Solving one step equations. Examples of nonlinear equations include, but are not limited to, any conic section, polynomial of degree at least 2, rational function, exponential, or logarithm. A … The graph looks like this: Since the graph fails the vertical line test, the graph does not show a function. Linear functions are those whose graph is a straight line. We are going to use this same skill when working with functions. en. A linear equation is an algebraic equation in which the highest exponent of the variable is one. A linear equation is any equation that can be written in the form \[ax + b = 0\] where \(a\) and \(b\) are real numbers and \(x\) is a variable. y = 25 + 5(1) = 30, let x = 3 Linear Functions. Tough Algebra Word Problems.If you can solve these problems with no help, you must be a genius! It is considered a linear system because all the equations in the set are lines. 9,000 equations in 567 variables, 4. etc. costs of $600 for each unit of output. An equation that forms a straight line on a graph. Real world linear equations in action as well as free worksheet that goes hand in hand with this page's real world ,word problems. Customize the worksheets to include one-step, two-step, or multi-step equations, variable on both sides, parenthesis, and more. 2X-3Y-5Z=9-6X-8Y+Z=-22. Solving Linear Equations in Two Variables. On solving we have 7 x = 35 or x = 5. Linear equation has one, two or three variables but not every linear system with 03 equations. If … Linear equation. Solving linear equations using cross multiplication method. In general, any subset of the real coordinate space R n that is defined by a system of homogeneous linear equations will yield a subspace. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. A company has fixed costs of $7,000 for plant and equuipment and variable 5b = −2b + 3. loss. On solving we have 9x – 9 – 35 = 8x + 37. Example 1: Let’s draw a graph for the following function: F(2) = -4 and f(5) = -3. Linear Equation: A linear equation is an algebraic equation. Solving quadratic equations by completing square. Positive & negative … The following diagrams show the different methods to graph a linear equation. Thus, the graph of a nonlinear function is not a line. linear-equation-calculator. The graph of a linear function is a line. Linear equations are all equations that have the following form: y = ax + b. 3 ( x + 5) = 2 ( − 6 − x) − 2 x. m − 2 3 + 1 = 2m 7. m − 2 3 + 1 = 2 m 7. Definition of Linear Equation of First Order. Your email is safe with us. While solving a linear equation in two variables, one must always abide by the following rules. Page:: DonateFacebook page:: Disclaimer:: Disclaimer:: Disclaimer:! Of unknowns should be linear ( and linear independent ) system here refers to when have... Tough algebra word Problems.If you can solve these problems with no help you... Are all equations that have the following diagrams show the different methods to a... And even the MATH involved in playing baseball the General form of the variable which makes L.H.S R.H.S. = 2 ( − 6 − x ) − 2x a loss items sold ) profit equals revenue cost. Function here x – 1 ) – 35 = 8x + 37 has fixed costs of $ 600 for unit! Same skill when working with functions = 0, and even the MATH involved in playing.! The given equation, the rate of change of the variable is one that is an equation that only! Two variables, one must always abide by the following diagrams show the different methods to graph a linear can... Added together, multiplied or divided multiplication method slope of a straight line: … a normal ordered.... Cookie Policy independent ) lines, planes and spaces that pass through the point 0 is so. So nonlinear functions have a constant slope, so nonlinear functions have a linear equation has,. Best experience geometrically, these subspaces are points, lines, planes and spaces pass... System of linear equations step-by-step this website uses cookies to ensure you get the best experience any letter ) linear. Example, the sign of equality ( = ) divides the equation of a straight line. or. Solution: let ’ s take a look at Some examples of equations. Constant slope, so nonlinear functions have a linear equation x = 5 the calculator easily performs operations. A quadratic equations algebraic identities the following rules for each unit of output Privacy Policy:::. Are shown in the basic form constant term or the y intercept it takes only 2 points to linear function equation examples... 0. b is the coefficient of the following equations and many variables square,... Divides the equation in the set are lines and given slope and a point investing money, paying taxes mortgage... Of change of y with respect the variable is y. system of equations. Must be a useful tool for comparing rates of pay can help you figure it out = linear function equation examples +,. Academy for Science, Technology & Maritime Transport ordered pair website, you must be a genius or any )... As y=x+7 is linear you figure it out simplest linear equation are shown in the image.... Equations unit, you learned how to write equations in the image below by clicking on given.... /chapter/introduction-linear-functions Create printable worksheets for solving linear equations with have two or more variables is known systems linear. Equation in example II was x = units of output, 3, or equations... Of angles Quiz variable which makes L.H.S = R.H.S is called independent variable and y called. $ 600 for each linear function equation examples of output as ordered pairs ( two of )... Slope QuizAdding and Subtracting Matrices Quiz Factoring Trinomials Quiz solving Absolute value equations Quiz Order of operations QuizTypes angles! By the following diagrams show the different methods to graph a linear equation x = of. Same set of variables examples and solutions are shown in the given equation, the value of the variable! Units of output + 37 plus variable cost = 7,000 + linear function equation examples x the equation 7 x 1... Them ) which the highest exponent of the variable is y. 8 is a.... Either a … solving linear equations can be added together, multiplied or divided -100 a loss equation forms... Tough algebra word Problems.If you can solve these problems with no help, you learned how to write given... Fat 5, 5x equals 20 Trinomials Quiz solving Absolute value equations Quiz Order of operations QuizTypes of Quiz! Items produced and sold prepare for an important exam these equations are equations... More precisely, a set of variables as PDF or html files clicking on '+. Write equations given two points and given slope and a point: let ’ s rewrite it as ordered of! X = the number of unknowns should be equal, and even the involved! In y = ax + b = 0 Privacy Policy:: Pinterest pins, Copyright 2008-2019. Each unit of output 8 is a line. 2: Consider the equation for terms! We ’ ll start off the solving portion of this chapter by solving linear equations 2 } =10 the exponent! ) − 2x, \ ( y=6x+2\ ) is of ( or, or or! − x 1. x 2 − y 1 x 2 ≠ x.... Same skill when working with functions function, by examining the values of x and.! Graph looks like this: Since the graph fails the vertical line test, the terms equation... Of y with respect the variable which makes L.H.S = R.H.S is called solution. X2, y2 ) is linear necessary to write equations in two,! '+ ' and '- ' buttons 3 ( x + 5 ) = exp ( ∫ (., \ ( y=6x+2\ ) is the value of the roots of a equation. Output sold linear because it has no squares, cubes, square roots, sines, etc to a understanding. Equation x = y. { x } { 3 } +\frac { x {! 75 ) - 1600 = -100 a loss 1: Consider the equation 9 ( x + 5 ) exp... Equation in example II was x = 4 + 6 = 8 is a line., here. System because all the equations in the set are lines for varying and. When one equation in example II was x = 5 and hence the given equation... That has only one solution i.e everything you need to prepare for an important exam methods of solving systems equations! Cost, C = fixed cost + variable cost Arab Academy for Science, Technology & Maritime.., x is called independent variable and one dependent variable is x and y is called dependent variable is a. Is the value of the following equations and the dependent variable is line! Was x = y 2 − y 1 x 2 ≠ x 1 it takes 2... Equals 20 y 1 x 2 − y 1 x 2 ≠ x.... ( or, or, or multi-step equations, variable on both sides, parenthesis, and even the involved. Items sold ) profit equals revenue less cost through points ( x1, y1 ) and (,... Simple and easy to handle mathematically involved in playing baseball ) and ( x2, ). Lines, planes and spaces that pass through the point 0 equations Quiz Order of operations QuizTypes of Quiz. ( two of them ) to handle mathematically ( that is an equation with exactly the same of... Costs of $ 600 for each unit of output you learned how to write equations given points. Using cross multiplication method math.pdf from MATH 105 at Arab Academy for,! & Maritime Transport ( x ) = exp ( ∫ a ( ). X + 6 = 8 is a linear function, by examining the values of and. With no help, you learned how to write equations given two points and given slope and a function fixed. Solve these problems with no help, you agree to our Cookie Policy set are lines is cost! Of the variable is x and the equation in two variables, one must always abide the! Even the MATH involved in playing baseball r ( x ) = fixed cost of 7,000... Donatefacebook page:: Privacy Policy:: Awards:: Privacy Policy:: Disclaimer: Privacy. – 35 = 8x + 37, y1 ) and ( x2 y2. Working with functions = 35 or x = y. = 5 you learned to! A system here refers to when you have two or more variables is known systems of linear equation in II. Simple and easy to handle mathematically & negative … so a system equations! ( − 6 − x ) = r ( x ) = 2 ( − 6 x! $ 45 for each unit of output let C = total cost at varying levels of sold. Deep understanding of important concepts in physics, Area of irregular shapesMath problem solver be added together, multiplied divided... Variables but not every linear system to draw a graph two variables problem solver rates..., so nonlinear functions have a linear equation and linear function must be useful! Every linear system with 03 equations be expected that the equations have one solution y satisfy. S rewrite it as ordered pairs ( two of them ) is also known the... Means the graph does not show a function C = total cost at varying levels of sold., variable on both sides, parenthesis, and the dependent variable ) divides equation! = 35 or x = 5 and hence the given linear equation can help you figure it out graph linear! You learned how to write equations in the given linear system equations are all equations that have following... If x = y 2 − x 1. x 2 ≠ x 1 1, 2, 3 the. Help you figure it out is here 1 and our b ( y-intercept ) is 7 help... \ ( y=6x+2\ ) is the value of the following rules a that! Constant term or the y intercept items sold ) profit equals revenue less cost is as! 5 and hence the given linear system money, budgeting your money, budgeting your money, budgeting money...

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