Linear Systems Word Problems

Math Word Problems with Solutions and Answers for Grade 10 Grade 10 math word problems with answers and solutions are presented. Solving linear equations form xa b c a the a. Example: A plane flying against the wind flew 270 miles in 3 hours. System Of. Systems of Linear Equations: Word Problems Jefferson Davis Learning Center, Sandra Peterson Use systems of linear equations to solve each word problem. Systems of Equations Word Problems Date_____ Period____ 1) Kristin spent $131 on shirts. Systems Quiz #1-5. (This worksheet emphasizes writing systems - you should already know how to solve them. Throughout history students have hated these. Watch Sal work through a harder Systems of linear equations word problem. Define the two variables (“let” statements) and write the two equations. Handout – Mathpower 10 (Pg 44 # 4, 9, 12, 14, 16). A word (or applied) problem involving unknown number (or quantity) can be translated into a linear equation consisting of one unknown. However, caution must be taken into account when review is repeatedly covered in your classroom. What are the numbers? (9. But word problems do not have to be the worst part of a math class. An objective function is a linear function in two or more variables that is to be optimized (maximized or minimized). y = sales job. Handout Solutions. Or, put in other words, we will now start looking at story problems or word problems. A Quadratic Equation is the equation of a parabola and has at least one variable squared (such as x 2) And together they form a System of a Linear and a Quadratic Equation. and i havent figured out the second one yet. The exactly dimension of Mesmerizing Algebra Word Problems Examples and solutions On Applications Of Linear Systems was 960x540 pixels. Find two consecutive odd integers. As you get further into Algebra 1, you will find that the real world problems become more complex. If the problem is not a story problem, skip to step 3. The most fundamental optimization problem tr eated in this book is the linear programming (LP) problem. The two word problems at the end of the review and in the assessment allows me to check student understanding of distinguishing between these 2 types of problems. Rodolfo has a total of 17 dimes and quarters worth $3. Algebra Need Answer To Math Problem 5th Grade Test. Simultaneous equations can help us solve many real-world problems. Use the MINVERSE function to return the inverse matrix of A. 50, and each student paid $1. , traveling at 70 mph to New York, which. How many nickels and how many dimes are there?. Word Problem Worksheet. If the problem is not a story problem, skip to step 3. November 13 - Systems Word Problems November 9 - Inequality Word Problems November 8 - Systems of Linear Inequalities November 7 - Graphing Linear Inequalities November 6 - Graphing Word Problems November 3 - SOE Word Problems Day 2 November 2 - Systems Word Problems November 1 - Unit 4 Systems Quiz October 31 - Unit 4 Quiz Review. Going from Words to a Linear. Fun math practice! Improve your skills with free problems in 'Solve a system of equations using elimination: word problems' and thousands of other practice. Solving linear systems word problems grade 10. Solving systems of linear equation word problems Home cleaning business plan problem solving techniques with children how to write a summary of a passage examples warehouse gym business plan ideas, recently published research papers 2017!. Kotter's Rentals rents a Cadillac Escalade for $99 for one week plus $0. The most fundamental optimization problem tr eated in this book is the linear programming (LP) problem. The best way to approach word problems is to "divide and conquer". A few important phrases are described below that can give us clues for how to set. The exactly dimension of Mesmerizing Algebra Word Problems Examples and solutions On Applications Of Linear Systems was 960x540 pixels. To solve a work word problem, multiply the hourly rate of the two people working together times the time spent working to get the total amount of time spent on the job. The fourth step, "take a. Mixture problems. Students will work through 4 clues to find the suspect, an accomplice, weapon and location. At a dance party, a group of boys and girls exchange dances as follows: one boy dances with 5 girls, a second boy second with 6 girls, and so on, the last boy dancing with all the girls. Expected Learning Outcomes The students will be able to: 1) Write a system of two linear equations to represent a word problem, then solve the system. 20 to buy the supplies to make a. Solve the following equations: x/2 - 1/2 = x/3 + 1/4, 2. You may not encounter these word problems a lot in algebra. You can work at most 20 hours next week. 1) IS 230 sold a total of 64 tickets to a basketball game. Knowing how to solve systems of equations is necessary to solve these types of problems. y = sales job. Example: The school that Matt goes to is selling tickets to a choral performance. This tutorial reviews systems of linear equations. How are you with solving word problems in Algebra? Are you ready to dive into the "real world" of inequalities? I know that solving word problems in Algebra is probably not your favorite, but there's no point in learning the skill if you don't apply it. Remember to show all 5 parts. Then we moved onto solving systems using the Substitution Method. The first order was for 13 bushes and 4 trees, and totalled $487. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. System of linear inequalities Graph of a system of linear inequalities Solution of a system of linear inequali… A set of two or more linear equati… A grocery bag contains x apples, each w… Which shows the graph of the solution s… Dominick is training for a race. 1) Find the value of two numbers if their sum is 12 and their difference is 4. Systems of Linear Equations Name MCAS Worksheet 1 Printed from myMCAS. Handout Solutions. Solutions of Systems of Linear Equations. Use a system of equations to solve problems. Define the two variables (“let” statements) and write the two equations. With this installment from Internet pedagogical superstar Salman Khan's series of free math tutorials, you'll learn how to work out linear function word problems. Well, in this lesson we’re going to make Solving Linear Equation Word Problems manageable with easy to follow tricks and steps. Chapter 7 : Systems of Linear Equations and Inequalities Is the availability of low-cost housing increasing or decreasing? How do you find the speed of the current in a river? In Chapter 7 you'll use linear systems and problem solving to help you answer these questions. Section 2: Canceling. In grade 10 you studied the point of intersection between two lines. Hamster Fun World charges an annual fee of $35 plus $1 for each day of hamster activities you attend. Fuel x costs $2 per gallon and fuel y costs $3 per gallon. Plus model problems explained step by step. Very commonly, system-of-equations word problems involves mixtures or combinations of some sort. First, select the range B6:D8. In many of the examples below, I'll use the whole equation approach. Grade 9, 10 and 11 Two Variable Linear Systems. Section 7-2 : Linear Systems with Three Variables. If you're seeing this message, it means we're having trouble loading external resources on our website. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. Then they use a decoder to get immediate feedback as they self-check the answers. Solving systems of linear equations algebraically is sometimes called the substitution method, but the process is the same no matter what it is called. Word problems solved by substitution Leslie and Gilbert drove a total of 447 miles in 11 hours. Watch Sal work through a harder Systems of linear equations word problem. 12 Represent and solve problems that can be modeled using a system of linear equations and/or inequalities in two variables, sketch the solution sets, and interpret the results within the context of the problem;. Housecleaning pays $5 per hour and your sales job pays $8 per hour. The first order was for 13 bushes and 4 trees, and totalled $487. System of linear equations solver This system of linear equations solver will help you solve any system of the form:. We refer to this property as the objective function of an LP problem. A Florist Must Make 5 Identical Bridesmaid Bouquets Systems Problem. How many of each type of shirt did he buy? 16) A class used cars and vans to go on a field. Glenna wants to rent a car for a trip to Key West for one week. Example: Find the solution to the following system of equations by graphing them. In the main program, all problems are automatically graded. Previous section Solving Systems of Linear Equations by Substitution Next section Solving Systems of Linear Equations by Addition/Subtraction Take a Study Break Literary Characters Summed Up in Quotes from The Office Sep 19, 2019. This Christmas Math Worksheet may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math. Dynamics (Force) problems ask you to relate motion to the forces causing it. Language:. y = gallons of fuel y. Systems of linear equations word problems worksheet. Solve Equations Word Problems Examples. Start studying Create a System of Linear Equations from Word Problems. Write the system of equations 3. The equation is created and solved in detail and can be used to check your own work. This is why much of our focus tends to be about teaching word problems creatively. S 8 6M Ia7dAeM qwrilt ghG MIonif ziin PiWtXe y qAtl 8g keFb 9ruaS T2P. To solve a work word problem, multiply the hourly rate of the two people working together times the time spent working to get the total amount of time spent on the job. Playlist Two-Variable Inequalities (Linear 5) Systems of Linear Inequalities. How To Solve basic linear function word problems in algebra. Word Problem #3 A salad contains ham and chicken. Bulut bought 5 footballs and 2 soccer balls and spent $140. - Using a system of equations to find the price of apples and oranges - Linear systems word problem with substitution method - System of equations to realize you are getting ripped off - Thinking about multiple solutions to a system Online Practice - Systems of equations word problems - Understanding systems of equations word problems Print Notes. Free worksheet(pdf) and answer key on the solving word problems based on linear equations and real world linear models. Three examples of using a system of linear equations to solve an application problem. SYSTEM of INEQUALITIES WORD PROBLEMS. Students, teachers, parents, and everyone can find solutions to their math problems instantly. Ah; so your curriculum didn't cover "uniform rate" word problems. 1 – Graphing Linear Relations; 1. Linear programming problems are applications of linear inequalities, which were covered in Section 1. Word Problems on Linear Equations : In this section, we will learn, how to solve word problems using linear equations. Solving Systems of Linear Equations by Graphing. Learn to apply linear equations to the real word with these money related problems. H ERE ARE SOME EXAMPLES of problems that lead to simultaneous equations. The first step in the formulation is to name the decision variables and their units of measurement unless the units of measurement are obvious. 9 – Problem. Interpret solutions in problem contexts. Which of the following systems of equations can be used to nd a, the number of adults attending,. Just begin by solving one of the equations for one of its variables. Attempting to solve these s. Section 7-1 : Linear Systems with Two Variables. Lots of word problems can be solved with systems of linear equations. Solving Linear Equation Word Problems 9 Terrific Examples. Systems of linear equations two variables a the math worksheet. How many of each coin does he have? The two variables used in this problem are d, number of dimes, and q, number of quarters. • To write systems of linear equations. The next day, you purchase 6 gal of paint and 2 brushes for $113. Pre-AP Algebra 2 Lesson 2-3 – Bellringer 2x2 Systems of Equations Word Problems 1) I am thinking of two numbers. 11 per mile over 100 miles. Solving Systems of Linear Equations by Graphing; Solving Systems of Linear Equations by Substitution; Solving Systems of Linear Equations by Addition; More Vocabulary; Word Problems and Lines; Solving Word Problems; Word Problems with Two Lines; Translating a Word Problem into a System of Equations. Write the problem by defining the objective function and the system of linear inequalities. Linear programming is a technique used to solve models with linear objective function and linear constraints. These are puzzles and riddles with numbers. That is where we get to answer the infamous question, when will we use this?. Solving a System of Equations. The first step in the formulation is to name the decision variables and their units of measurement unless the units of measurement are obvious. 50 per hour and your job as a car wash attendant pays $6 per hour. Word Problem #1. Consider the following problem: Peter and Nadia like to race each other. Housecleaning pays $5 per hour and your sales job pays $8 per hour. In these disciplines, problem solving is part of a larger process that encompasses problem determination, de-duplication, analysis, diagnosis, repair, and other steps. Solving a Linear Programming Problem. All test items have been released to the public by the Massachusetts Department of Elementary and Secondary Education. System of inequalities word problems worksheet worksheets for all download and share free on bonlacfoods com. Two such. Systems of Linear Equations Section WILA What is Linear Algebra? C10 (Robert Beezer) In Example TMP the rst table lists the cost (per kilogram) to manufacture each of the three varieties of trail mix (bulk, standard, fancy). Let's jump straight to an example. Examples of systems present at center 1: Examples of word problems at center 2: To differentiate the lesson, for example for remedial students, this version of word problems could be used instead:. Linear Function Word Problems. Here's the instructions. Linear Equations Word Problems. Question 1 On the set of axes below, graph ( )=1 2 −3and ( )={2 +1, <0 2− 2, ≥0. Homework Solutions. Previous section Solving Systems of Linear Equations by Substitution Next section Solving Systems of Linear Equations by Addition/Subtraction Take a Study Break Literary Characters Summed Up in Quotes from The Office Sep 19, 2019. This FREE worksheet on Systems of Equations contains 15 Christmas and Winter themed word problems which students need to solve by writing and solving systems of equations. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Word Problems Involving Systems of Linear Equations. Linear Systems Word Problems. A nut mixture contains 20% peanuts by mass and another contains 40%. He put part of it in a term deposit that paid 4% per annum, and the remainder in a treasury bill that paid 5% per annum. Visual, handout or organizer of key words for slope-intercept form of linear equation word problems. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Section 2: Canceling. Know more about what are linear equations in two variables, its examples, questions, word problems and applications by visiting BYJU'S. Word Problems; Problem Solving Strategy; Example. Solving word problems like this one aren't so bad if you know what to do. A system of linear equations is any system whose equations only contain constant or linear terms. (Respond to what is asked for. There Has to Be a System for This Sweet Problem. Use the MINVERSE function to return the inverse matrix of A. The e-mail states that 24 individuals placed, earning a combined total of 53 points. 11) 9 less than a number is at most that same number divided by 2. Solving word problems using linear systems of two equations in two unknowns In this lesson we present some typical word problems and show how to solve them using linear systems of two equations in two unknowns. Math Highlights Students work with a system of linear equations that has two equations in two variables. Systems of Equations Word Problems Date_____ Period____ 1) Find the value of two numbers if their sum is 12 and their difference is 4. Applications of Linear Systems In this video lesson we will learn about some Applications of Linear Systems and Linear Models in Business, Science and Engineering. Also, students will practice writing equations from stories or graphs. Ah; so your curriculum didn't cover "uniform rate" word problems. Students will use knowledge of solving systems of linear functions using three methods; substitution, linear combination (or addition) and graphing. 00 per day to care each cat. Graphs to Linear Equations - Sample Math Practice Problems The math problems below can be generated by MathScore. Some of these types are so popular, they are categorized under specific headings. Two Step Equations Word Problems Practice Khan Academy. Attempting to solve these s. Linear Systems Word Problems. Juan has 2 styles of expenditures that entire to seventy six. A summary of Solving Systems of Linear Equations by Addition/Subtraction in 's Systems of Equations. The activity gives students a real life perspective on. Section 7-2 : Linear Systems with Three Variables. Improve your math knowledge with free questions in "Solve a system of equations by graphing: word problems" and thousands of other math skills. In order to join an online learning community, there is a $20 startup fee. b Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. com - Antonella Artuso. By this point in the year students will have solved many Algebraic word problems. 17) A farmhouse shelters 15 animals. Depending on your preferred input format, you can create equations in Word in either one of UnicodeMath or LaTeX formats by selecting the format from the Equations tab. Language:. 5 points in a major chess tournament. The graph of the intersection appears in the second screen. Grade 8 Algebra Word Problems The following are some examples and solutions for algebra word problems that you will commonly encounter in grade 8. Write a system of equations describing the following word problem: The Lopez family had a rectangular garden with a 20 foot perimeter. Question; Assign variables to the unknown quantities; Set up the equations; Rearrange equation \((1)\) and substitute into equation \((2)\) Calculate the number of tricycles \(t\) Write the final answer; Example. L Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Systems of Equations Word Problems Date_____ Period____ 1) Find the value of two numbers if their sum is 12 and their difference is 4. A nut mixture contains 20% peanuts by mass and another contains 40%. So, to find the values of x and y, add lines together in a certain way. Not a member? Join Now. In Mathematics, we apply our skills and understanding to solve problems in our lives. A cleanser containing 20% ammonia is mixed with another cleanser containing 10% ammonia to dilute its strength. Improve your math knowledge with free questions in "Solve a system of equations by graphing: word problems" and thousands of other math skills. Define the variables. Focus: Using algebra to analyze and solve problems in context, otherwise known as word problems. Dynamics (Force) problems ask you to relate motion to the forces causing it. Students will work through 4 clues to find the suspect, an accomplice, weapon and location. Many word problems will give rise to systems of equations --- that is, a pair of equations like this: You can solve a system of equations in various ways. Linear Programming Word Problems. The bills do not list the per-item price. Problem sets for each method include one solution, no solution, infinitely many solutions and word problems. Writing Expression for Geometric Sequences. Kotter’s Rentals rents a Cadillac Escalade for $99 for one week plus $0. A Florist Must Make 5 Identical Bridesmaid Bouquets Systems Problem. Unit 2 - Analytic Geometry. This Algebra Worksheet may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math. They enlarged their garden to be twice as long and. Example of a table of contents for a research paper examples determinism vs free will essays college board essay writing tips, 5 steps to essay writing haitian revolution essays, it assessment spreadsheet good research paper hook business plan sample on word solving boundary value problems for ordinary differential equations in matlab with. 1) Green Day wants to record a demo. answers pdfFREE PDF. Using the accompanying set of axes, graph the. Systems of Equations (WORD PROBLEMS) Systems of Inequalities;. They begin the Overview by setting up an analysis of two different cell phone plans. When both the numerator and denominator are decreased by 6, then the denominator becomes 12 times the numerator and determine the fraction. LINEAR EQUATIONS. Together, they made 208 paintings, and there were 3 times as many baby rhinos as adult rhinos. Linear Systems Word Problems Emporium Mathematics; 4 videos; 338 views; Last updated on Sep 9, 2018; Systems of Equations: Word Problems - Mixtures by Emporium Mathematics. com is a free math website that explains math in a simple way, and includes lots of examples, from Counting through Calculus. Some of the worksheets for this concept are Systems of equations word problems, Systems word problems, Linear systems word problems date, Systems of linear equations in three variables, Linear equations, Practice solving systems of equations 3 different, Systems word problems, Real. Use a system of equations to solve problems. The school took in $171 on the second day by selling 3 senior citizen tickets and 14 child tickets. Their sum is 13. Glenna wants to rent a car for a trip to Key West for one week. We need to talk about applications to linear equations. The best method to use depends on the forms of the given equations and how precise the solution should be. You must choose a method before you solve a word problem. The video tutorial shows how to write and solve a linear system from a set of given information. Algebra II » Linear Systems and Matrices » Solving Systems of Linear Equations in Three Variables. To solve this system of linear equations in Excel, execute the following steps. Once students solve the linear system problems and write their answers as ordered pairs, they need to plot their solutions on t. Fun math practice! Improve your skills with free problems in 'Solve a system of equations using elimination: word problems' and thousands of other practice. ) Do at least one example with a. They have more questions to be answered and require more steps to find the solution. Some word problems require the use of systems of linear equations. The best way for you to learn to solve word problems is to practice to translating the words into math problems. Linear Equations in Three Variables To solve a systems of linear equations in three variables x , y , and z , we need to use the following steps: 1) Rewrite the systems of equations in three variables as systems of linear equations in two variables. System of linear inequalities Graph of a system of linear inequalities Solution of a system of linear inequali… A set of two or more linear equati… A grocery bag contains x apples, each w… Which shows the graph of the solution s… Dominick is training for a race. These Word Problems Worksheets are a good resource for students in the 5th Grade, 6th. I help you understand how to set up Linear Systems of Equations from word problems. Using the clues given in the problem we. November 13 - Systems Word Problems November 9 - Inequality Word Problems November 8 - Systems of Linear Inequalities November 7 - Graphing Linear Inequalities November 6 - Graphing Word Problems November 3 - SOE Word Problems Day 2 November 2 - Systems Word Problems November 1 - Unit 4 Systems Quiz October 31 - Unit 4 Quiz Review. Solve equations of these forms fluently. Now that you can do these difficult algebra problems, you can trick your friends by doing some fancy word problems; these are a lot of fun. Our company offers free of charge innovative publishing worksheets to support educators offer college students any head-start inside of a class. pdf: File Size: 979 kb: File Type: pdf. Sample Problems From Intermediate Algebra Sample problems are under the links in the "Sample Problems" column and the corresponding review material is under the "Concepts" column. How many cones were sold? 2. The best way to approach word problems is to "divide and conquer". The solution of a linear system is the ordered pair that is a solution to all equations in the system. Solving a linear equation usually means finding the value of y for a given value of x. Solving linear systems of equations by the elimination method: 0 1 2 Graphing linear systems of inequalities: 0 1 2 Defining variables and setting up a system of equations to represent a word problem: 0 1. Define variables 2. 3 Story problems involving systems of linear equations in two variables. com is a free math website that explains math in a simple way, and includes lots of examples, from Counting through Calculus. A collection of coins contains only nickels and dimes. Solving systems of equations word problems worksheet worksheets for all download and share free on bonlacfoods c. Word Problems Leading to Inequalities. Write one complete sentence that explains what the solution (x,y) means in context to the word problem. 1) In basketball you score 2 points for a field goal and 1 point for a free throw. How To Solve basic linear function word problems in algebra. In other words, the solution is the point where the two lines intersect. Systems of linear equations word problems worksheet. Basically I have a linear system of equations, where every coefficient is a variable. Basic to advanced level. Section 7-2 : Linear Systems with Three Variables This is going to be a fairly short section in the sense that it’s really only going to consist of a couple of examples to illustrate how to take the methods from the previous section and use them to solve a linear system with three equations and three variables. When a linear equation has two variables, as it usually does, it has an infinite number of solutions. Systems of Equations and Inequalities Systems of two linear inequalities Systems of two equations Systems of two equations, word problems Points in three dimensions Planes Systems of three equations, elimination Systems of three equations, substitution Cramer's rule:2x2,3x3. Check your answer. 5 Solving Systems Using Inverse Matrices 231 SOLUTION OF A LINEAR SYSTEM Let AX= Brepresent a system of linear equations. A system of 3 equations can be used to solve a problem where there are three unknowns and enough information to make three equations. Solving Linear Equations - Age Problems Objective: Solve age problems by creating and solving a linear equa-tion. Question; Assign variables to the unknown quantities; Set up a system of equations. Solving linear systems word problems grade 10 Example of a school philosophy essay examples Who am i essay examples for college math extended essay tips newspaper business plan template printable biography essay samples start up construction business plan, printing business plan philippines phenomenology dissertation education. Section 2: Canceling. Simply put, two-step equations - word problems are two step equations expressed using words instead of just numbers and mathematical symbols. 9%, and his federal income tax rate 14%. Linear-Quadratic Word Problems Choose any two of the questions below and answer them on a separate piece of paper. In order to join a dancing club, there is a $30 startup fee and a $4 monthly fee. However, caution must be taken into account when review is repeatedly covered in your classroom. Linear motion (also called rectilinear motion) is a one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. Problems of Solutions of Systems of Linear Equations. In this section we are going to be looking at non-linear systems of equations. Previous section Solving Systems of Linear Equations by Substitution Next section Solving Systems of Linear Equations by Addition/Subtraction Take a Study Break Literary Characters Summed Up in Quotes from The Office Sep 19, 2019. They will create a system of linear equations and they must be able to solve the systems of linear equations by the three methods: graphing, substitution, and elimination. System of Linear Equations. Solving linear programming word problems worksheet the elegant essay writing lessons by lesha myers microeconomics research paper topic ideas management plan examples in business plan sample assign static ip to raspberry pi 3. In this video, we solve a problem about a toy factory. Solving linear systems word problems grade 10 du 18 mars au 24 février • Meurthe-et-Moselle Grand Débat national : participez aux 2 réunions d’initiatives locales à vocation économique le 28 février et le 5 mars. In the main program, all problems are automatically graded. When digging into the earth, the temperature rises according to the following linear equation: t = 15 + 0. Unicode math. A system of 3 equations can be used to solve a problem where there are three unknowns and enough information to make three equations. A word (or applied) problem involving unknown number (or quantity) can be translated into a linear equation consisting of one unknown. Some of the worksheets for this concept are Systems of equations word problems, Systems word problems, Linear systems word problems date, Systems of linear equations in three variables, Linear equations, Practice solving systems of equations 3 different, Systems word problems, Real. Common Core Math Worksheets - High School. The term problem solving is used in numerous disciplines, sometimes with different perspectives, visuals, and often with different terminologies. Solve Equations Word Problems Examples. Admission prices are $6 for adults and $4 for children. A linear objective function is to be maximized or minimized. Precalculus word problem A rectangular piece of cardboard measuring 12 inches by 18 inches is to be made into a box w/ an open top by cutting equal-sized squares from each corner and folding up the sides. The system is consistent and dependent Example Line 1: y= x +2 Line 2: y= 3x + 6 If you divide line 2 by 3, It will be exactly the same as line 1 The graph of the two lines are exactly the same.