Conic Sections: Parabola and Focus. In the next few videos, we're What is the solution of this system of linear equations? So 3 comma 3 satisfies The procedure to use the graphing linear inequalities calculator is as follows: Step 1: Enter the inequality equation in the respective input field Step 2: Now click the button "Graph" to get the graph Step 3: Finally, the graph of linear inequality will be displayed in the new window What is Meant by the Graphing Linear Inequalities? Graphing a system of linear equations is as simple as graphing two straight lines. The technical storage or access is required to create user profiles to send advertising, or to track the user on a website or across several websites for similar marketing purposes. To graph a line from a slope-intercept equation, take the value of the slope and put it over 1. Example: Using a graph, find the solution for the equations y = 2x + 7 and y = -3x - 8. mx plus b form, or slope-intercept form. It appears we may have a connection issue. A solution is an ordered pair that corresponds to a point where the two lines in the rectangular coordinate plane intersect. Direct variation. It turns out that this is not always the case. Direct link to Ricardo Longoria's post if the question is This is a dependent system, and the infinitely many solutions are expressed using the form \((x, mx+b)\). How to Graph the Equation in Algebra Calculator First go to the Algebra Calculator main page. Take one line and identify two points on And it looks like I intersect In this article, we shall discuss linear programming, calculating and . Graphical Method Calculator - Linear Programming Objective: Objective Function: X + X Constraints Constraint 1: X + X Constraint 2: X + X X 1, X 2 0 + Graph Reset Members-Only Content Calculation of the intersections with the axes to graph each constraint. At PM Calculators we are working to bring you the best tools gathered in one place. 3 times 2 is 6, minus 6 is 0. To solve this system of linear equations in Excel, execute the following steps. Yes, your guess right: you have infinite intersection points, which means that you have infinite solutions. Type another linear equation (Ex: y = 2x + 3, 3x - 2y = 3 + 2/3 x, etc.) set to this equation, all of the coordinates many unknowns. intersection of those lines. So the equation, the line bunch, you're going to move down that same bunch. Well, we can do the This is called the y-intercept form, and it's probably the easiest form to use to graph linear equations. equal to 5, you go to the line, and you're going to see, Mathematics is a way of dealing with tasks that involves numbers and equations. Geometrically, solutions are the points where the graphs intersect. Get immediate feedback and guidance with step-by-step solutions and Wolfram Problem Generator. Let's see if x is equal to In slope-intercept form, we can easily see that the system consists of two lines with the same slope and different \(y\)-intercepts. Mathematics is a way of dealing with tasks that involves numbers and equations. It is a good practice to always check your solutions. Linear Equations with Fractions Calculator To provide the best experiences, we use technologies like cookies to store and/or access device information. To place an order, please fill out the form below. Direct link to Seed Something's post The video shows 5:00 grap, Posted 6 months ago. The rule is clear: when there is no intersection between the lines, there is no solution to the system. to negative 6. And let's see if it satisfies \(\left\{\begin{aligned}x+y&=6\\ 5x+2y&=2 \end{aligned}\right.\). So in this situation, this the line. as exact as possible. In this section, we limit our study to systems of two linear equations with two variables. But pretty awesome app, excellent and abundantly helpful. lines are equal, then we have infinite solutions. Can some one tell me what section I need to do do be up to speed. Enter your queries using plain English. right 1, you're going to move down 1. System of two linear equations. Steps. When we say system of equations, Requested URL: byjus.com/graphing-linear-equations-calculator/, User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10_15_7) AppleWebKit/605.1.15 (KHTML, like Gecko) Version/15.5 Safari/605.1.15. I need to put my answers in the following format: I am assuming that they are two vectors, which one has a scalar s. Could you help me out in solving this? What about this line? Remember that (2,5) is an (x,y) coordinate where x=2 and y=5. A system of equations is a set of one or more equations involving a number of variables. Students begin by using their calculator to graph 9 linear systems. Such two-by-two systems often appear when solving word problems, proportion problems and assignment problems with constraint. Or click the example. The difference between two numbers is \(12\) and their sum is \(4\). Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. intersection of the lines. And we have a slope of 1, The technical storage or access is necessary for the legitimate purpose of storing preferences that are not requested by the subscriber or user. on both lines, or essentially, a point of Example Problem. Solve linear systems using the graphing method. Other resources may express this set using set notation, \(\{(x, y) | y=\frac{2}{3}x3\}\), which reads the set of all ordered pairs \((x, y)\) such that \(y\) equals two-thirds \(x\) minus \(3\). Sometimes the lines do not cross and there is no point of intersection. There is no simultaneous solution, \(\). So in this case, the first one You can solve systems of equations by graphing using the following steps: For each, equation graph the line. $$ Thank you for your help! Well, you can use substitution or elimination. Identify dependent and inconsistent systems. If any of these terms are perplexing, don't worry. The system is said to be inconsistent otherwise, having no solutions. have several equations. To solve a system of linear equations using a graph on the TI-84 Plus C Silver Edition, follow the example below. Likewise, you must also select the sign of the inequalities. And let's say the other equation For a new problem, you will need to begin a new live expert session. A) 2 y = 4 x + 2 B) 2 y = x + 7 Systems of Linear Equations Worksheets What is a System? Well, you look at it here, it's These are called the solutions to a system of equations. Solving Systems of Linear Equations by Graphing. the y-intercept. Finding slope from two points. When x is 0 here, 0 plus Solve the new equation. One instrument that can be used is Graphically solving a system of linear equations calculator. a line, every point on this line is a solution And the system is dependent. I spoke with my team and we will make note of this for future training. Consenting to these technologies will allow us to process data such as browsing behavior or unique IDs on this site. If you want to enhance your academic performance, start by setting realistic goals. (I got this one wrong) Which equation can pair with x + 2y = 5 to create an inconsistent system? So maybe when you take x is We need some way to express the solution sets to dependent systems, since these systems have infinitely many solutions, or points of intersection. that satisfy y is equal to x plus 3. Each of them constrain And remember, when I'm drawing Is there a different problem you would like further assistance with? The solutions to systems of equations are the variable mappings such that all component equations are satisfiedin other words, the locations at which all of these equations intersect. Solving optimization exercises with the graphical method will be easier with our graphical method calculator for linear programming problems. Here, \(y\) depends on \(x\). Since many students cannot adequately understand how the graphs were generated, we have developed a version with detailed step-by-step explanations of the solution of the problem. see more algebraic ways of solving these than drawing their Well, think about it. Once you have found the key details, you will be able to work out what the problem is and how to solve it. This point lies on both lines. Interactive, free online graphing calculator from GeoGebra: graph functions, plot data, drag sliders, and much more! video shows how to solve systems of linear equations on a TI83 or TI84 series graphing calculator. You must enter the coefficients of the objective function and the constraints. { "4.01:_Solving_Linear_Systems_by_Graphing" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.02:_Solving_Linear_Systems_by_Substitution" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.03:_Solving_Linear_Systems_by_Elimination" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.04:_Applications_of_Linear_Systems" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.05:_Solving_Systems_of_Linear_Inequalities_(Two_Variables)" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "4.0E:_4.E:_Review_Exercises_and_Sample_Exam" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Real_Numbers_and_Their_Operations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "02:_Linear_Equations_and_Inequalities" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "03:_Graphing_Lines" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "04:_Solving_Linear_Systems" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "05:_Polynomials_and_Their_Operations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "06:_Factoring_and_Solving_by_Factoring" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "07:_Rational_Expressions_and_Equations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "08:_Radical_Expressions_and_Equations" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "09:_Solving_Quadratic_Equations_and_Graphing_Parabolas" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Appendix_-_Geometric_Figures" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "license:ccbyncsa", "showtoc:no", "authorname:anonymous", "licenseversion:30", "program:hidden" ], https://math.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fmath.libretexts.org%2FBookshelves%2FAlgebra%2FBeginning_Algebra%2F04%253A_Solving_Linear_Systems%2F4.01%253A_Solving_Linear_Systems_by_Graphing, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), 4.2: Solving Linear Systems by Substitution, status page at https://status.libretexts.org. Every point on this line Solution: Step 1: Rewrite the linear equations in slope-intercept form. if the number before x is positive than the line looks like this /. And you use each equation as a Systems of linear equations are very commonly found in different context of Algebra. This page titled 4.1: Solving Linear Systems by Graphing is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous. \(\begin{array}{c|c} {-2x+5y=-15}&{-4x+10y=10}\\{-2x+5y\color{Cerulean}{+2x}\color{black}{=-15}\color{Cerulean}{+2x}}&{-4x+10y\color{Cerulean}{+4x}\color{black}{=10}\color{Cerulean}{+4x}}\\{5y=2x-15}&{10y=4x+10}\\{\frac{5y}{\color{Cerulean}{5}}\color{black}{=\frac{2x-15}{\color{Cerulean}{5}}}}&{\frac{10y}{\color{Cerulean}{10}}\color{black}{=\frac{4x+10}{\color{Cerulean}{10}}}}\\{y=\frac{2}{5}x-3}&{y=\frac{2}{5}x+1} \end{array}\). You will need to get assistance from your school if you are having problems entering the answers into your online assignment. 1, 2, 3, 4, 5, 6. So that coordinate pair, or that The x and y coordinates of the intersection will be the solution. More general systems involving nonlinear functions are possible as well. Graphing linear inequalities. TI-84 Plus CE Graphing Calculator For Dummies, 3rd Edition Explore Book Buy On Amazon Matrices are the perfect tool for solving systems of equations (the larger the better). Geometrically, a linear system consists of two lines, where a solution is a point of intersection. Graph your problem using the following steps: Type in your equation like y=2x+1. Experts will give you an answer in real-time, using the square root method to solve quadratic equations, can you take the log of both sides of an equation, calculate area of triangle in c programming. Explanation of the area to shade depending on the type of inequality. ), Type another linear equation (Ex: y = 2x + 3, 3x - 2y = 3 + 2/3 x, etc.). y = -3 When graphing the lines, take care to choose a good scale and use a straightedge to draw the line through the points; accuracy is very important here. equation right there. You can use math to determine all sorts of things, like how much money you'll need to save for a rainy day. intersection. If you click on "Tap to view steps" you will see the steps are now numbered. This video shows how to use the Ti-84 Graphing Calculator to solve a system of linear equations in 2 variables graphically. everything that satisfies this purple equation is on the So it's going to look In this blog post,. The graphing method is not the most accurate method for determining solutions, particularly when the solutions have coordinates that are not integers. When the lines are graphed, the solution will be the (x,y) ordered pair where ` 5.1: Solve Systems of . Determine the price at which the quantity demanded is equal to the quantity supplied. So, how many intersection If you want to get the best homework answers, you need to ask the right questions. \(\left\{\begin{aligned} xy&=1 \\ 2x+3y&=5 \end{aligned}\right.\). I will end the session - please reconnect if you still need assistance. Where on the graph of \(5x+2y=30\) does the \(x\)-coordinate equal the \(y\)-coordinate? Let me write that down. 14/10 would reccomend :p, also, when your answer isn't the same as the app it still exsplains how to get the right answer. Graph the lines and determine the point of intersection. Arc Length S = R Gur Emir mausoleum - mu How Many Radians? Solving System of Equations by Substitution method is the easiest way to solve linear equations calculations. Let's do another one. So every time you move So this line is going For Example, Solving x-3y=4, x+2y=1 linear equations using substitution we get x=11/5, y=-3/5. 1) Press [Y=] to access the Y= editor. Direct link to Hannah's post How do you graph an equat, Posted a year ago. Try MathPapa Algebra Calculator About MathPapa Back to System of Equations Calculator So, the methodology is simple: You start with a linear system, and the first thing you do is to graph the two You get 3 is equal to negative SW solve systems of linear and quadratic equations. So we may express all the ordered pair solutions \((x, y)\) in the form \((x, mx+b)\), where \(x\) is any real number. In slope-intercept form, we can easily see that the system consists of two lines with the same slope and same \(y\)-intercept. The most commonly found systems in basic Algebra courses For example, if the slope was 5, the slope would be 5/1. the bottom equation. If you're looking for a punctual person, you can always count on me. We are more than happy to answer any math specific question you may have about this problem. Next, insert the MINVERSE function shown below. The graphing method consists of representing each of the linear equations as a line on a graph. Use a graphing calculator to solve the following system of linear equations: {eq}y=0.15x -0.12, \ 2.5x - y=- 2.4 {/eq}. Use the graph to answer the following questions. Graphing calculators and computers are able to solve systems of equations, eliminating the need for time-consuming computations. Uh oh! Pro Tip: Online graphing calculators like Desmos can help you check your work quickly and easily. Discover Resources Midpoint Quad Activity SM1 dew_unitcircle centroid Andragradsprov uppgift 9 When graphing linear equations, it helps if the equations are written in slope-intercept form, y=mx+b, where m is the slope and b is the y-intercept. is y is equal to negative x plus 6. So we were able to solve this And then 1, 2, 3, 4, both of these lines. Solve 2x + 3 = 4(x - 6) - 2 using the graphing function of the TI-83/84 series calculators. Determination of special cases such as unbounded, unbounded or infeasible solutions. Our online calculator based on wolfram alpha system can solve more complex systems of inequalities than the one just discussed. Let's say we have y is (If you have a second equation use a semicolon like y=2x+1 ; y=x+3) Press. Expert tutors are available to help with any subject. Graphing a system of linear equations by graphing the system on a graphing calculator and using the intersect function. And this is already in You can contact support with any questions regarding your current subscription. While we cover a very wide range of problems, we are currently unable to assist with this specific problem. Check out our Math Homework Helper for tips and tricks on how to tackle those tricky math problems. More Examples At 0 comma 3. something like this. \(\left\{\begin{aligned}2x+3y&=9 \\ 4x6y&=18\end{aligned}\right.\). I am only able to help with one math problem per session. Use Math24.pro for solving differential equations of any type here and now. So the point 0, 3 is on Systems of linear equations involving more than two variables work similarly, having either one solution, no solutions or infinite solutions (the latter in the case that all component equations are equivalent). points do you have? Solving system graphically calculator How to Graph the Equation in Algebra Calculator. Often you'll see an equation that looks like this: y = 1/4x + 5, where 1/4 is m and 5 is b. Mathway requires javascript and a modern browser. Or if you move to the right a Our examples of problem solving will help you understand how to enter data and get the correct answer. you can write the equation of the line in slope-intercept form. Let's say I have the equation my hand-drawn graphs, so maybe it's Step 1: Make sure the linear equations are in the form of y = mx . Solve equations and systems of equations with Wolfram|Alpha, A powerful tool for finding solutions to systems of equations and constraints, Partial Fraction Decomposition Calculator. They don't have to be, but they Therefore, they are parallel and will never intersect. equal to 3x minus 6. Which problem would you like to work on? solving a system of equations graphically - To solve a system of linear equations graphically we graph both equations in the same coordinate system. Solutions Graphing Practice; New Geometry Equations Inequalities System of Equations System of Inequalities Basic . The steps for solving linear systems using the graphing method are outlined in the following example. Free System of Inequalities calculator - Graph system of inequalities and find intersections step-by-step. not the exact-- let's check this answer. The calculator first will try to get the lines into slope-intercept and will provide you with a graph and with an So if we check it into the first gee, when x is equal to 5 on that line, y is equal Doing homework can help you learn and understand the material covered in class. However, with a little bit of practice, anyone can learn to solve them. If the lines do not intersect (they are parallel), then the system of equations has no solution. Get calculation help online Get math help online by chatting with a tutor or watching a video lesson. There will be more accurate algebraic methods in sections to come, but for now, the goal is to understand the geometry involved when solving systems. Functions: What They Are and How to Deal with Them, Normal Probability Calculator for Sampling Distributions, find the intersection Learn about linear equations using our free math solver with step-by-step solutions. linear equations. Step 3: Use the graph to estimate the point where the lines intersect and check to see if it solves the original system. we just mean many equations that have In this section, we will study linear systems consisting of two linear equations each with two variables. Determine whether the given ordered pair is a solution to the given system. This calculator solves Systems of Linear Equations using Gaussian Elimination Method, Inverse Matrix Method, or Cramer's rule.Also you can compute a number of solutions in a system of linear equations (analyse the compatibility) using Rouch-Capelli theorem.. The first step to finding the solution to this system of equations is to graph both lines as follows: Notice that the ONLY intersection point for this system of equations is at (2,5). 1. To solve the equation by the graphing calculator, you need to enter the given linear equation in the input field and then click on the calculate/solve button to display the graph of the given linear equations in just a few seconds. graphing systems of linear equations calculator - Free graphing calculator instantly graphs your math problems. That is why we include a series of online resources, where linear programming is a must. Please make sure you are in the correct subject. another equation. 3-- 1, 2, 3. Since \((1, 0)\) does not satisfy both equations, it is not a solution. Sometimes systems consist of two linear equations that are equivalent. x, y pair, must satisfy both equations. They are, in fact, the same line. Not consenting or withdrawing consent, may adversely affect certain features and functions. Direct link to Shreyas's post Slope is measured as Rise, Posted 10 years ago. Solving systems of linear equations. A solution to a linear system, or simultaneous solution, to a linear system is an ordered pair \((x, y)\) that solves both of the equations. 2x + 3y = 16.9 5x = y + 7.4 What is the solution to the system of linear equations? You may speak with a member of our customer support team by calling 1-800-876-1799. This way, you can see which lines are parallel and . two graphs and trying to find their intersection Then, we need to find the intersection example y is equal to negative x plus 3. to both of these equations. Direct link to Dyamine8's post How do we know that X's s, Posted 11 years ago. And I want to graph all of the approximation. Direct link to Patrick's post y = -x + 3 Well You're in luck! In this example, it is important to notice that the two lines have the same slope and same \(y\)-intercept. 3, y equals 3 definitely satisfies both these it looks like we're at 1, 2, 3 comma 1, 2, 3. Solving Systems of Equations by Substitution Steps So, the steps for using the substitution method to solve a system of linear equations are: Rewrite one of the equations to isolate one variable. Remember, How to graph your problem. To determine a math equation, one would need to first understand the problem at hand and then use mathematical operations to solve it. Graphing lines using standard form. Note: the formula bar indicates that the cells contain an array formula. So our line will look something like that right there. \((3, 2); \left\{\begin{aligned} x+y&=-1\\-2x-2y&=2 \end{aligned}\right.\), \((5, 0); \left\{\begin{aligned} x+y&=1\\2x2y&=2 \end{aligned}\right.\), \((2, 6); \left\{\begin{aligned}x+y&=4\\3xy&=12 \end{aligned}\right.\), \((2, 7); \left\{\begin{aligned} 3x+2y&=8\\5x3y&=11 \end{aligned}\right.\), \((0, 3); \left\{\begin{aligned}5x5y&=15\\13x+2y&=6 \end{aligned}\right.\), \((12, 14); \left\{\begin{aligned} x+y&=14\\2x4y&=0 \end{aligned}\right.\), \((\frac{3}{4}, \frac{1}{4}); \left\{\begin{aligned} xy&=1\\4x8y&=5 \end{aligned}\right.\), \((3, 4); \left\{\begin{aligned} \frac{1}{3}x+\frac{1}{2}y&=1 \\ \frac{2}{3}x\frac{3}{2}y&=8 \end{aligned}\right.\), \((5, 3); \left\{\begin{aligned} y&=35x10 \\ y&=5 \end{aligned}\right.\), \((4, 2); \left\{\begin{aligned} x&=47\\x+4y&=8 \end{aligned}\right.\), \(\left\{\begin{aligned} y &=\frac{3}{2}x + 6\\y&=x + 1 \end{aligned}\right.\), \(\left\{\begin{aligned} y& =\frac{3}{4}x + 2\\y&=\frac{1}{4}x 2 \end{aligned}\right.\), \(\left\{\begin{aligned} y& =x 4\\y&=x + 2 \end{aligned}\right.\), \(\left\{\begin{aligned} y&= 5 x + 4\\y& = 4 x 5 \end{aligned}\right.\), \(\left\{\begin{aligned} y& =\frac{2}{5} x + 1\\ y& =\frac{3}{5} x \end{aligned}\right.\), \(\left\{\begin{aligned}y&=\frac{2}{5} x + 6\\y& =\frac{2}{5} x +10 \end{aligned}\right.\), \(\left\{\begin{aligned} y&= 2\\y& =x + 1 \end{aligned}\right.\), \(\left\{\begin{aligned} y& = 3\\ x&= 3 \end{aligned}\right.\), \(\left\{\begin{aligned} y& = 0\\y& =\frac{2}{5} x 4 \end{aligned}\right.\), \(\left\{\begin{aligned} x& = 2 \\y& = 3 x \end{aligned}\right.\), \(\left\{\begin{aligned} y& =\frac{3}{5} x 6\\y& =\frac{3}{5} x 3 \end{aligned}\right.\), \(\left\{\begin{aligned} y&=\frac{1}{2}x + 1\\ y&=\frac{1}{2}x + 1 \end{aligned}\right.\), \(\left\{\begin{aligned}2 x + 3 y &=18 \\ 6 x + 3 y&= 6 \end{aligned}\right.\), \(\left\{\begin{aligned} 3 x + 4y &=20\\2 x + 8y &= 8\end{aligned}\right.\), \(\left\{\begin{aligned} 2 x +y &=1 \\2 x 3 y& = 9\end{aligned}\right.\), \(\left\{\begin{aligned} x + 2y&=8\\5 x + 4y&= 4\end{aligned}\right.\), \(\left\{\begin{aligned}4 x + 6y &=36\\2 x 3 y &= 6\end{aligned}\right.\), \(\left\{\begin{aligned} 2 x 3 y &=18\\6 x 3 y&= 6\end{aligned}\right.\), \(\left\{\begin{aligned} 3 x + 5y &=30\\ 6 x 10y&=10\end{aligned}\right.\), \(\left\{\begin{aligned}x + 3 y &=3\\5 x 15 y&=15\end{aligned}\right.\), \(\left\{\begin{aligned}x y &= 0\\ x +y &= 0\end{aligned}\right.\), \(\left\{\begin{aligned}y &=x\\y x& = 1\end{aligned}\right.\), \(\left\{\begin{aligned}3 x + 2y &= 0\\ x &= 2\end{aligned}\right.\), \(\left\{\begin{aligned}2 x +\frac{1}{3}y &=\frac{2}{3}\\ 3 x +12y&= 2\end{aligned}\right.\), \(\left\{\begin{aligned}\frac{1}{10}x +\frac{1}{5}y &= 2\\ \frac{1}{5} x +\frac{1}{5}y&= 1\end{aligned}\right.\), \(\left\{\begin{aligned}\frac{1}{3} x \frac{1}{2}y &= 1 \\ \frac{1}{3} x +\frac{1}{5}y& = 1 \end{aligned}\right.\), \(\left\{\begin{aligned} \frac{1}{9}x +\frac{1}{6}y &= 0 \\ \frac{1}{9}x +\frac{1}{4}y &=\frac{1}{2}\end{aligned}\right.\), \(\left\{\begin{aligned} \frac{5}{16}x \frac{1}{2}y &= 5\\ \frac{5}{16}x +\frac{1}{2}y &=\frac{5}{2} \end{aligned}\right.\), \(\left\{\begin{aligned} \frac{1}{6}x\frac{1}{2}y&=\frac{9}{2} \\ \frac{1}{18}x+\frac{1}{6}y&=\frac{3}{2} \end{aligned}\right.\), \(\left\{\begin{aligned} \frac{1}{2}x\frac{1}{4}y&=\frac{1}{2} \\ \frac{1}{3}x\frac{1}{2}y&=3\end{aligned}\right.\), \(\left\{\begin{aligned} y&=4\\x&=5 \end{aligned}\right.\), \(\left\{\begin{aligned} y&=3\\x&=2\end{aligned}\right.\), \(\left\{\begin{aligned}y&=0\\x&=0\end{aligned}\right.\), \(\left\{\begin{aligned}y&=2\\y&=3\end{aligned}\right.\), \(\left\{\begin{aligned}y&=5\\y&=5\end{aligned}\right.\), \(\left\{\begin{aligned}y&=2\\y2&=0\end{aligned}\right.\), \(\left\{\begin{aligned}x&=5\\x&=1\end{aligned}\right.\), \(\left\{\begin{aligned}y&=x\\x&=0\end{aligned}\right.\), \(\left\{\begin{aligned}4x+6y&=3\\x+y&=2\end{aligned}\right.\), \(\left\{\begin{aligned}2x+20y&=20\\3x+10y&=10\end{aligned}\right.\). 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