https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:introduction-to-systems-of-equations/v/trolls-tolls-and-systems-of-equations 2021-06-21T15:29:05Z Systems of equations: trolls, tolls (1 of 2) A troll forces us to use algebra to figure out the make-up of his currency. We end up setting up a system of equations. Sal Khan video https://cdn.kastatic.org/googleusercontent/b_y-BaRBvyK-maEDUf9uN9H7WjoV7DwXUfAETMvkfjJYUdZeB1r2U_xSikLLt_09JSlBT8wDB1tLwzz-_xBCEP85ow Systems of equations: trolls, tolls (1 of 2) A troll forces us to use algebra to figure out the make-up of his currency. We end up setting up a system of equations. https://cdn.kastatic.org/ka-youtube-converted/OcNt-36QKu8.mp4/OcNt-36QKu8.mp4 411 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:introduction-to-systems-of-equations/v/solving-the-troll-riddle-visually 2021-06-21T15:29:11Z Systems of equations: trolls, tolls (2 of 2) Solving the system of equations visually. Now we can save the prince/princess. Sal Khan video https://cdn.kastatic.org/googleusercontent/trvex1O3M0Avapk25H-0NxxrsDToa-gOX76bhvFhq4Tb-C4VFHOO8hCRBHnfw7Bf6Y7YMD1zxKey0TbzPQS7ehomkQ Systems of equations: trolls, tolls (2 of 2) Solving the system of equations visually. Now we can save the prince/princess. https://cdn.kastatic.org/ka-youtube-converted/H-HfmyUzPw8.mp4/H-HfmyUzPw8.mp4 341 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:introduction-to-systems-of-equations/v/testing-a-solution-for-a-system-of-equations 2021-06-21T15:29:05Z Testing a solution to a system of equations Sal checks whether (-1,7) is a solution of the system: x+2y=13 and 3x-y=-11. Sal Khan video https://cdn.kastatic.org/googleusercontent/ckn5jlapHG75xl41VkQ-rEF2GRxD5zhraMy6wCs9H1Ufc9_VjKJQZXB3_fqXXWr1XNBBhgFQ5ChWuKPxFMZn-IIurQ Testing a solution to a system of equations Sal checks whether (-1,7) is a solution of the system: x+2y=13 and 3x-y=-11. https://cdn.kastatic.org/ka-youtube-converted/SkMNREAMNvc.mp4/SkMNREAMNvc.mp4 158 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:introduction-to-systems-of-equations/v/solving-systems-graphically 2021-06-21T15:29:10Z Systems of equations with graphing: y=7/5x-5 & y=3/5x-1 Sal graphs the following system of equations and solves it by looking for the intersection point: y=7/5x-5 and y=3/5x-1. Sal Khan video https://cdn.kastatic.org/googleusercontent/jmKPgEP2ePAnbt52yGyOU-3SO5b0FTZiIJ9xeqsoitZHmJHEx_DNwpk-oA-IZbx9WgPCCVer_d1DHhya9DjixZHm Systems of equations with graphing: y=7/5x-5 & y=3/5x-1 Sal graphs the following system of equations and solves it by looking for the intersection point: y=7/5x-5 and y=3/5x-1. https://cdn.kastatic.org/ka-youtube-converted/VhH2nEDCd68.mp4/VhH2nEDCd68.mp4 142 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:introduction-to-systems-of-equations/v/solving-systems-graphically-examples 2022-10-06T11:39:29.972393653Z Systems of equations with graphing: exact & approximate solutions Sal solves a system of two linear equations in standard form, and then approximates the solution of a system whose solution isn't clearly visible. video https://cdn.kastatic.org/ka_thumbnails_cache/e69bfee3-b4ab-4f54-bad1-e74465d359d8_1280_720_base.png Systems of equations with graphing: exact & approximate solutions Sal solves a system of two linear equations in standard form, and then approximates the solution of a system whose solution isn't clearly visible. https://cdn.kastatic.org/ka-youtube-converted/o4pbAQgJYjI.mp4/o4pbAQgJYjI.mp4 356 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:introduction-to-systems-of-equations/e/graphing_systems_of_equations 2026-05-04T16:10:46.51963925Z Systems of equations with graphing Graph a system of equations in slope-intercept or standard form, and find their solution using the graphs. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:number-of-solutions-to-systems-of-equations/v/solving-systems-by-graphing-2 2021-02-07T17:14:01Z Number of solutions to a system of equations Sal is given three lines on the coordinate plane, and identifies one system of two lines that has a single solution, and one system that has no solution. Sal Khan video https://cdn.kastatic.org/googleusercontent/7FrrO5V3Z35wNeM9yk7cry7zxf3JyBChiv2KVJF_no0lzKa_--k2D84y5juX_y4V3wyJdD8rWlgas49_O-d9nGR4 Number of solutions to a system of equations Sal is given three lines on the coordinate plane, and identifies one system of two lines that has a single solution, and one system that has no solution. https://cdn.kastatic.org/ka-youtube-converted/BNHLzEv6Mjg.mp4/BNHLzEv6Mjg.mp4 206 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:number-of-solutions-to-systems-of-equations/v/practice-thinking-about-number-of-solutions-to-systems 2021-02-07T17:14:01Z Number of solutions to a system of equations graphically Sal determines how many solutions the following system of equations has by considering its graph: 10x-2y=4 and 10x-2y=16. Sal Khan video https://cdn.kastatic.org/googleusercontent/LSPenQW1-1XWojpiiFK3oJpAy4UchBLSv3EjgjiP-nsEaT6i02-ntLjopewu763fNrdsLTi7lLULqHO0tqp9oWy08g Number of solutions to a system of equations graphically Sal determines how many solutions the following system of equations has by considering its graph: 10x-2y=4 and 10x-2y=16. https://cdn.kastatic.org/ka-youtube-converted/SuB1gkto9LU.mp4/SuB1gkto9LU.mp4 396 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:number-of-solutions-to-systems-of-equations/e/graphical-solutions-to-systems 2026-04-07T23:36:17.218592739Z Number of solutions to a system of equations graphically Determine the number of solutions of a given system of equations by considering its graph. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:number-of-solutions-to-systems-of-equations/v/consistent-and-inconsistent-systems 2021-02-07T17:14:00Z Solutions to systems of equations: consistent vs. inconsistent A consistent system of equations has at least one solution, and an inconsistent system has no solution. Watch an example of analyzing a system to see if it's consistent or inconsistent. Sal Khan video https://cdn.kastatic.org/googleusercontent/8Fb8sxtm5whcUxsxn_mAVRisE6QYnd4vD680aGtnCgw-86az6xrM0KEI9O1flgbIEoJuyyPBlx0jmHzrX4D-oduxNQ Solutions to systems of equations: consistent vs. inconsistent A consistent system of equations has at least one solution, and an inconsistent system has no solution. Watch an example of analyzing a system to see if it's consistent or inconsistent. https://cdn.kastatic.org/ka-youtube-converted/Ix8Nne-a-KQ.mp4/Ix8Nne-a-KQ.mp4 329 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:number-of-solutions-to-systems-of-equations/v/independent-and-dependent-systems 2021-02-07T17:14:00Z Solutions to systems of equations: dependent vs. independent A dependent system of equations has infinite solutions, and an independent system has a single solution. Watch an example of analyzing a system to see if it's dependent or independent. Sal Khan video https://cdn.kastatic.org/googleusercontent/CSBiMO9wDE9k94CGv-KYG-WMMK-3FmKltaNip2fcR_uoROqtO2HY1FhwwjPyNXa4Rgnm9qgwVvZ8mU6hSOzwlgg Solutions to systems of equations: dependent vs. independent A dependent system of equations has infinite solutions, and an independent system has a single solution. Watch an example of analyzing a system to see if it's dependent or independent. https://cdn.kastatic.org/ka-youtube-converted/WSpF5uvApLA.mp4/WSpF5uvApLA.mp4 310 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:number-of-solutions-to-systems-of-equations/v/examples-algebraically-analyzing-solutions-to-systems 2022-10-06T11:39:29.972393653Z Number of solutions to a system of equations algebraically Sal solves several examples where he reasons about the number of solutions of systems of equations using algebraic reasoning. video https://cdn.kastatic.org/ka_thumbnails_cache/518c64a9-c141-4ad9-a0da-40161c1df7ef_1280_720_base.png Number of solutions to a system of equations algebraically Sal solves several examples where he reasons about the number of solutions of systems of equations using algebraic reasoning. https://cdn.kastatic.org/ka-youtube-converted/ogEddosP1G8.mp4/ogEddosP1G8.mp4 361 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:number-of-solutions-to-systems-of-equations/e/constructing-consistent-and-inconsistent-system 2026-03-10T04:30:37.966653692Z Number of solutions to a system of equations algebraically Determine the number of solutions of a given system of equations by considering its algebraic solution process. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:number-of-solutions-to-systems-of-equations/v/understanding-systems-of-equations-example-2 2021-02-07T17:14:01Z How many solutions does a system of linear equations have if there are at least two? Sal answers this question for you! Sal Khan video https://cdn.kastatic.org/googleusercontent/OaMLsNwtdOD-hgjco9UrJWOy10jjG5yKCvjx2OCWYTrqt7pt6dvourgZoX_bsEhOc73rHmeBev_qB1v5DsPvVKIHeg How many solutions does a system of linear equations have if there are at least two? Sal answers this question for you! https://cdn.kastatic.org/ka-youtube-converted/bzseeIVEslA.mp4/bzseeIVEslA.mp4 132 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:number-of-solutions-to-systems-of-equations/a/number-of-solutions-to-system-of-equations-review 2022-04-27T20:03:03.137387157Z Number of solutions to system of equations review A system of linear equations usually has a single solution, but sometimes it can have no solution (parallel lines) or infinite solutions (same line). This article reviews all three cases. article https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:solving-systems-of-equations-with-substitution/v/practice-using-substitution-for-systems 2021-06-21T15:29:06Z Systems of equations with substitution: -3x-4y=-2 & y=2x-5 Learn to solve the system of equations -3x - 4y = -2 and y = 2x - 5 using substitution. Sal Khan video https://cdn.kastatic.org/googleusercontent/628RjCJ_CsSqK0HqbIq8aNzgQ257wowuvmMVKKnUJmO9mpgtI2jeA2M8Zz15za_g4gB2lK1zv-A2gIIc8JNmvrKv Systems of equations with substitution: -3x-4y=-2 & y=2x-5 Learn to solve the system of equations -3x - 4y = -2 and y = 2x - 5 using substitution. https://cdn.kastatic.org/ka-youtube-converted/GWZKz4F9hWM.mp4/GWZKz4F9hWM.mp4 261 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:solving-systems-of-equations-with-substitution/e/systems_of_equations_with_substitution 2025-12-28T12:11:43.202987517Z Systems of equations with substitution Solve systems of equations where one of the equations is solved for one of the variables. Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:solving-systems-of-equations-with-substitution/a/substitution-method-review-systems-of-equations 2021-06-21T15:29:14Z Substitution method review (systems of equations) The substitution method is a technique for solving a system of equations. This article reviews the technique with multiple examples and some practice problems for you to try on your own. article https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:solving-systems-elimination/v/simple-elimination-practice 2021-04-19T14:43:37Z Systems of equations with elimination: x-4y=-18 & -x+3y=11 Sal solves the following system of equations by eliminating x: x-4y=-18 and -x+3y=11. Sal Khan video https://cdn.kastatic.org/googleusercontent/d2DLGZYUCnN2rbaWWNiLKf7sMsIOEzLeMxjvB8VePoc29qHG-cvvnfdD6Agj2ILs1j3E1MCGU3OGfG7dOql900w Systems of equations with elimination: x-4y=-18 & -x+3y=11 Sal solves the following system of equations by eliminating x: x-4y=-18 and -x+3y=11. https://cdn.kastatic.org/ka-youtube-converted/NPXTkj75-AM.mp4/NPXTkj75-AM.mp4 359 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:solving-systems-elimination/e/systems_of_equations_with_elimination_0.5 2025-12-28T12:11:43.202987517Z Systems of equations with elimination Solve systems of equation with one-step elimination (e.g., x-values or y-values cancel each other out). Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:solving-systems-elimination/v/solving-systems-of-equations-by-multiplication 2021-05-24T11:10:24Z Systems of equations with elimination (and manipulation) In some cases, we need to slightly manipulate a system of equations before we can solve it using the elimination method. See how it's done in this video. Sal Khan video https://cdn.kastatic.org/googleusercontent/8fSMiND4KoKRtNsTeuNYD3FNP5jjdekKoO6VEN3OYxMZXYCyDO4Mt-TrR7tmWEco3yi9_9rFgkM_27-e2vEw7aEl Systems of equations with elimination (and manipulation) In some cases, we need to slightly manipulate a system of equations before we can solve it using the elimination method. See how it's done in this video. https://cdn.kastatic.org/ka-youtube-converted/wYrxKGt_bLg.mp4/wYrxKGt_bLg.mp4 721 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:solving-systems-elimination/e/systems_of_equations_with_elimination 2025-12-28T12:11:43.202987517Z Systems of equations with elimination challenge Solve systems of equation with multi-step elimination (e.g., a manipulation is needed in order for x-values or y-values to cancel each other out). Tomer Gal exercise https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:equivalent-systems-equations/v/why-we-do-the-same-thing-to-both-sides-basic-systems 2021-04-19T14:43:38Z Why can we subtract one equation from the other in a system of equations? The example of a scale where we try to achieve balance helps to explain why we can subtract one equation from the other in a system of equations. Sal Khan video https://cdn.kastatic.org/googleusercontent/bpyrDg2UujNzFGs5RS2uaywsBlXjte_td8DC7eXQTiDA9EMK9CvnV7tGhyyxaJSMPwGNIQVgoc5tx5PFisdwO7xi Why can we subtract one equation from the other in a system of equations? The example of a scale where we try to achieve balance helps to explain why we can subtract one equation from the other in a system of equations. https://cdn.kastatic.org/ka-youtube-converted/h9ZgZimXn2Q.mp4/h9ZgZimXn2Q.mp4 493 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:solving-systems-elimination/a/elimination-method-review 2021-04-29T07:49:27Z Elimination method review (systems of linear equations) The elimination method is a technique for solving systems of linear equations. This article reviews the technique with examples and even gives you a chance to try the method yourself. article https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:number-of-solutions-to-systems-of-equations/v/inconsistent-systems-of-equations 2021-04-19T14:43:38Z Systems of equations number of solutions: fruit prices (1 of 2) Sal gives an example of a system of equations that has no solution! Sal Khan video https://cdn.kastatic.org/googleusercontent/966TVCMSq2me01GOkpmJfVXTQtwRJBxVxCQeO5g8zMv4tw21hkn2EfjHob5oOaIT56_1uH3LWADXXJiZsiLTp4FzMg Systems of equations number of solutions: fruit prices (1 of 2) Sal gives an example of a system of equations that has no solution! https://cdn.kastatic.org/ka-youtube-converted/NJRyxmommqQ.mp4/NJRyxmommqQ.mp4 565 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:number-of-solutions-to-systems-of-equations/v/infinite-solutions-to-systems 2021-04-19T14:43:38Z Systems of equations number of solutions: fruit prices (2 of 2) Sal gives an example of a system of equations that has infinite solutions! Sal Khan video https://cdn.kastatic.org/googleusercontent/N3hZWxNmgG-G_cqbpr7m9LnntVXeMW4hSiLmCD3_Pu2PPU6IDGv59T9kdl6BDHZLDCFIaXRsTSlwKY13SZQyuuMM9A Systems of equations number of solutions: fruit prices (2 of 2) Sal gives an example of a system of equations that has infinite solutions! https://cdn.kastatic.org/ka-youtube-converted/MpRUEGD2-_c.mp4/MpRUEGD2-_c.mp4 428 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:solving-systems-of-equations-with-substitution/v/solving-systems-with-substitution 2021-04-19T14:43:37Z Systems of equations with substitution: potato chips To solve a system of equations using substitution... 1. Isolate one of the variables in one of the equations, e.g. rewrite 2x+y=3 as y=3-2x. 2. You can now express the isolated variable using the other one. *Substitute* that expression into the second equation, e.g. rewrite x+2y=5 as x+2(3-2x)=5. 3. Now you have an equation with one variable! Solve it, and use what you got to find the other variable. Sal Khan video https://cdn.kastatic.org/googleusercontent/WFPUyv4Z65vD5DDKfQDmETlkFwVvjIApANcxoJ1brGA5Ov6_-mJYCK1T3gdsSAdyFhQ3yT4i4PDoXd77JMOaBo2a Systems of equations with substitution: potato chips To solve a system of equations using substitution... 1. Isolate one of the variables in one of the equations, e.g. rewrite 2x+y=3 as y=3-2x. 2. You can now express the isolated variable using the other one. *Substitute* that expression into the second equation, e.g. rewrite x+2y=5 as x+2(3-2x)=5. 3. Now you have an equation with one variable! Solve it, and use what you got to find the other variable. https://cdn.kastatic.org/ka-youtube-converted/2EwPpga_XPw.mp4/2EwPpga_XPw.mp4 321 Systems of linear equations in two variables https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:solving-systems-elimination/v/king-s-cupcakes-solving-systems-by-elimination 2021-04-19T14:43:38Z Systems of equations with elimination: King's cupcakes Sal uses simple elimination to figure out how many cupcakes are eaten by children and adults. Sal Khan video https://cdn.kastatic.org/googleusercontent/uMqqyuzKoKwjhXs4XtirvI9Tvum_GYku3_fY8Qft5Som0zvncK75ujNlYBfP36Y0YUX3i8RkXurmzGljaP-WZWw Systems of equations with elimination: King's cupcakes Sal uses simple elimination to figure out how many cupcakes are eaten by children and adults. https://cdn.kastatic.org/ka-youtube-converted/xCIHAjsZCE0.mp4/xCIHAjsZCE0.mp4 540 Systems of linear equations in two variables