Simultaneous equations - Vedic Maths Training Classes Pune - Bhushan 9370571465
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The document solves three systems of simultaneous equations. The first system is x + 2y = 10 and 2x + 3y = 18, which is solved to get x = 6 and y = 2. The second system is x + 3y = 11 and 2x + 3y = 13, which is solved to get x = 2 and y = 3. The third system is 4x + 5y = 14 and 3x + 3y = 9, which is solved to get x = 1 and y = 2.
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Gr 11 equations
Quadratic equations can be solved in several ways:
1) Factorizing, by finding two numbers whose product is the constant term and sum is the coefficient of the x term.
2) Using the quadratic formula.
3) Substitution, by letting an expression like x^2 + 2x equal a variable k, and solving the simplified equation for k and back substituting.
4) Squaring both sides, but this can introduce extraneous solutions so one must check solutions.
Maths lesson1
The document discusses methods for solving systems of linear equations in two variables, specifically the elimination method. It provides examples of using the elimination method to solve sample systems of linear equations. The key steps are: 1) rewriting the equations so coefficients of the variable being eliminated are opposite, 2) adding/subtracting the equations to eliminate one variable, 3) solving the resulting equation for the eliminated variable, and 4) substituting back into one of the original equations to solve for the other variable. Five sample systems are provided and the reader is prompted to try solving them using the elimination method.
Tugas 1
1. The document is a math worksheet containing word problems to solve quadratic equations. It asks students to solve equations by factoring, completing the square, and using the quadratic formula.
2. Some example equations included x2 + 5x - 50 = 0, 2x2 + 3x + 1 = 0, and x2 - 15x + 30 = 0.
3. The last question asks students to determine the value of P such that the equation (p + 3)x3 + 3x - 4 = 0 has two equal roots.
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Multiplying Polynomials
1) The document discusses multiplying different types of polynomials, including: monomials, binomials, and polynomials with more than two terms.
2) It provides examples and steps for multiplying a monomial by a monomial, monomial by a polynomial, binomial by binomial, and a polynomial by a polynomial with three or more terms.
3) The key steps are distributing terms, applying rules like the distributive property, and combining like terms in the results.
Tugas mtk soal 2
The document provides examples of solving quadratic equations by factoring, completing the square, and using the quadratic formula. It asks to:
1) Solve quadratic equations by factoring.
2) Solve quadratic equations by completing the square.
3) Solve quadratic equations using the quadratic formula.
4) Determine the value of p that makes a given quadratic equation have equal roots.
Ruffini
Ruffini's rule is a method for dividing a polynomial P(x) by a binomial x-a. The document provides two examples of using Ruffini's rule to divide polynomials. It involves writing the coefficients of the polynomials, then multiplying and subtracting terms of the divisor from the dividend according to the rule to obtain the quotient and remainder.
Simultaneous equations (2)
The document discusses solving systems of 3 linear equations with 3 unknowns. It provides examples of using the elimination method, which involves rewriting the system as two smaller systems, eliminating the same variable from each, solving the resulting system of 2 equations for the remaining 2 variables, then substituting back into one of the original equations to find the third variable. The solution is written as an ordered triple (x, y, z). It demonstrates this process on examples and encourages practicing this method.
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Gr 11 equations
Quadratic equations can be solved in several ways:
1) Factorizing, by finding two numbers whose product is the constant term and sum is the coefficient of the x term.
2) Using the quadratic formula.
3) Substitution, by letting an expression like x^2 + 2x equal a variable k, and solving the simplified equation for k and back substituting.
4) Squaring both sides, but this can introduce extraneous solutions so one must check solutions.
Maths lesson1
The document discusses methods for solving systems of linear equations in two variables, specifically the elimination method. It provides examples of using the elimination method to solve sample systems of linear equations. The key steps are: 1) rewriting the equations so coefficients of the variable being eliminated are opposite, 2) adding/subtracting the equations to eliminate one variable, 3) solving the resulting equation for the eliminated variable, and 4) substituting back into one of the original equations to solve for the other variable. Five sample systems are provided and the reader is prompted to try solving them using the elimination method.
Tugas 1
1. The document is a math worksheet containing word problems to solve quadratic equations. It asks students to solve equations by factoring, completing the square, and using the quadratic formula.
2. Some example equations included x2 + 5x - 50 = 0, 2x2 + 3x + 1 = 0, and x2 - 15x + 30 = 0.
3. The last question asks students to determine the value of P such that the equation (p + 3)x3 + 3x - 4 = 0 has two equal roots.
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Multiplying Polynomials
1) The document discusses multiplying different types of polynomials, including: monomials, binomials, and polynomials with more than two terms.
2) It provides examples and steps for multiplying a monomial by a monomial, monomial by a polynomial, binomial by binomial, and a polynomial by a polynomial with three or more terms.
3) The key steps are distributing terms, applying rules like the distributive property, and combining like terms in the results.
Tugas mtk soal 2
The document provides examples of solving quadratic equations by factoring, completing the square, and using the quadratic formula. It asks to:
1) Solve quadratic equations by factoring.
2) Solve quadratic equations by completing the square.
3) Solve quadratic equations using the quadratic formula.
4) Determine the value of p that makes a given quadratic equation have equal roots.
Ruffini
Ruffini's rule is a method for dividing a polynomial P(x) by a binomial x-a. The document provides two examples of using Ruffini's rule to divide polynomials. It involves writing the coefficients of the polynomials, then multiplying and subtracting terms of the divisor from the dividend according to the rule to obtain the quotient and remainder.
Simultaneous equations (2)
The document discusses solving systems of 3 linear equations with 3 unknowns. It provides examples of using the elimination method, which involves rewriting the system as two smaller systems, eliminating the same variable from each, solving the resulting system of 2 equations for the remaining 2 variables, then substituting back into one of the original equations to find the third variable. The solution is written as an ordered triple (x, y, z). It demonstrates this process on examples and encourages practicing this method.
Polinomials division
This document demonstrates how to divide two polynomials. It shows the step-by-step process of dividing the polynomial x3 + 2x2 - 4x + 3 by the polynomial x2 - 1. The quotient is x + 2 and the remainder is -3x + 5.
Algebra
The document provides examples and explanations of operations involving polynomials, including:
1) Adding polynomials by combining like terms such as 5x + 3x and finding the sum of polynomials using tiles.
2) Performing addition, subtraction, and multiplication of polynomials with various terms.
3) Dividing polynomials using tiles to represent the division operation and finding quotients and remainders.
Zeros of a polynomial function
The document discusses polynomial functions and their roots. It begins by defining that the roots of a polynomial function are the values of x that make the function equal to 0. It then provides examples of finding the roots of linear and quadratic equations. Next, it introduces the Rational Root Theorem, which states that possible rational roots must be factors of the constant term and leading coefficient. Examples are given to demonstrate applying the theorem. The document concludes by using synthetic division to find all three roots of a cubic polynomial given one known root.
Tugas mtk soal 2
1. The document provides solutions to several quadratic equations using factoring, completing the square, and the quadratic formula.
2. The equations are solved step-by-step showing the work. Common solutions included factoring the equation into two binomial factors and setting each factor equal to zero, or using the quadratic formula.
3. Additional context is provided about solving quadratic equations with different methods like completing the square or using the standard formula.
Simultaneous equations
The document discusses solving simultaneous equations. It provides examples of simultaneous equations involving two variables (x and y) and two equations, including one linear and one non-linear equation. Methods for solving the simultaneous equations include expressing one variable in terms of the other, substituting one equation into the other, and solving for the variables. Solutions may have multiple answer pairs for x and y.
Addition and subtraction of polynomials
The document provides information about adding and subtracting algebraic expressions:
- Like terms are algebraic expressions with the same variables and exponents.
- Unlike terms cannot be combined.
- To add algebraic expressions, combine like terms by adding the coefficients.
- To subtract expressions, change the sign of the second expression and then add as if adding.
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The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key points are eliminating variables to reduce the equations, then solving the reduced equations to find the values of the variables.
Polynomials
This document provides information about polynomials, including their standard form, degree, leading term, and the factor theorem. The degree of a polynomial is the highest exponent among its terms, and the leading term is the term with the highest degree and its coefficient. Examples demonstrate how
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This document discusses solving quadratic equations by factoring. It begins by defining a quadratic equation in standard form and explaining the zero factor property. It then provides examples of solving quadratic equations through factoring and setting each factor equal to zero. Finally, it demonstrates solving word problems by setting up and solving the resulting quadratic equation.
Zeros or roots of a polynomial if a greater than1
The document discusses finding the zeros or roots of polynomial functions using the Rational Root Theorem. It provides three examples of applying the Rational Root Theorem and synthetic division to determine the actual roots of given polynomial functions. The key steps are to find the possible rational roots using the Rational Root Theorem, then use synthetic division or factoring to determine which possible roots are the actual roots that make the polynomial function equal to zero.
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Factorising
The document provides examples of factorizing algebraic expressions. It begins with expanding simple expressions like 3g + 4 and finding the common factors of terms like 12 and 20. The next section provides more examples of factorizing expressions like x2 + 2x by finding the highest common factor. It emphasizes factorizing completely, like writing 3abc(ab4 + 9) instead of 9abc(ab4 + 9). The document concludes with an exercise for students to practice factorizing expressions and an extension with more challenging examples. It provides the opportunity for a 10 minute practice session to reinforce the steps of factorizing algebraic expressions.
Sketching the graph of a polynomial function
This document discusses how to find the x-intercepts and y-intercepts of a polynomial function. To find the x-intercepts, one must factor the polynomial completely and set each factor equal to zero to solve for x. To find the y-intercept, set x equal to zero and simplify the polynomial. For the example polynomial function x^4 - 5x^2 + 4, the x-intercepts are (1,0), (-1,0), (2,0), (-2,0) and the y-intercept is (0,4).
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Extracting the roots
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Factoring
The document discusses the zero factor theorem and provides examples of using it to solve quadratic equations. The zero factor theorem states that if p and q are algebraic expressions, then pq = 0 if and only if p = 0 or q = 0. This means a quadratic equation can be solved by factoring it into two linear factors and setting each factor equal to zero. Five examples are provided that show factoring quadratic equations, applying the zero factor theorem to set the factors equal to zero, and solving for the roots.
Multiplication of polynomials
The document discusses multiplying polynomials, including multiplying monomials, combining like terms, and special cases such as the sum and difference of binomials, squares of binomials, and cubes of binomials. Examples are provided for multiplying polynomials with 2, 3, or 4 terms. Formulas and step-by-step workings are shown for finding products of binomial expressions.
Factorising quadratic expressions 1
This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
Solving quadratic equation using completing the square
1. The document provides steps for solving quadratic equations using completing the square. It involves isolating terms with x, finding the square of half the coefficient of x, completing the square of the left side of the equation, applying the square root property, and solving for the roots.
2. Three examples are shown applying these steps: finding the solution set of x^2 - 18x = -17 to be x = 17 and x = 1; solving x^2 + 10x - 11 = 0 to get x = 1 and x = -11; and solving x^2 = -24 + 10x to get x = 6 and x = 4.
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The document provides examples and explanations of operations involving polynomials, including:
1) Adding polynomials by combining like terms such as 5x + 3x and finding the sum of polynomials using tiles.
2) Performing addition, subtraction, and multiplication of polynomials with various terms.
3) Dividing polynomials using tiles to represent the division operation and finding quotients and remainders.
Zeros of a polynomial function
The document discusses polynomial functions and their roots. It begins by defining that the roots of a polynomial function are the values of x that make the function equal to 0. It then provides examples of finding the roots of linear and quadratic equations. Next, it introduces the Rational Root Theorem, which states that possible rational roots must be factors of the constant term and leading coefficient. Examples are given to demonstrate applying the theorem. The document concludes by using synthetic division to find all three roots of a cubic polynomial given one known root.
Tugas mtk soal 2
1. The document provides solutions to several quadratic equations using factoring, completing the square, and the quadratic formula.
2. The equations are solved step-by-step showing the work. Common solutions included factoring the equation into two binomial factors and setting each factor equal to zero, or using the quadratic formula.
3. Additional context is provided about solving quadratic equations with different methods like completing the square or using the standard formula.
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The document discusses solving simultaneous equations. It provides examples of simultaneous equations involving two variables (x and y) and two equations, including one linear and one non-linear equation. Methods for solving the simultaneous equations include expressing one variable in terms of the other, substituting one equation into the other, and solving for the variables. Solutions may have multiple answer pairs for x and y.
Addition and subtraction of polynomials
The document provides information about adding and subtracting algebraic expressions:
- Like terms are algebraic expressions with the same variables and exponents.
- Unlike terms cannot be combined.
- To add algebraic expressions, combine like terms by adding the coefficients.
- To subtract expressions, change the sign of the second expression and then add as if adding.
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The document describes how to solve simultaneous equations using three steps: 1) eliminate a variable from the equations, 2) solve for the remaining variable, and 3) substitute back to find the eliminated variable. It provides an example showing these steps to solve two simultaneous equations for x and y. It then shows another example involving eliminating variables to create two new equations that can be solved simultaneously for y and z. The key points are eliminating variables to reduce the equations, then solving the reduced equations to find the values of the variables.
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Factorising
The document provides examples of factorizing algebraic expressions. It begins with expanding simple expressions like 3g + 4 and finding the common factors of terms like 12 and 20. The next section provides more examples of factorizing expressions like x2 + 2x by finding the highest common factor. It emphasizes factorizing completely, like writing 3abc(ab4 + 9) instead of 9abc(ab4 + 9). The document concludes with an exercise for students to practice factorizing expressions and an extension with more challenging examples. It provides the opportunity for a 10 minute practice session to reinforce the steps of factorizing algebraic expressions.
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This document discusses how to find the x-intercepts and y-intercepts of a polynomial function. To find the x-intercepts, one must factor the polynomial completely and set each factor equal to zero to solve for x. To find the y-intercept, set x equal to zero and simplify the polynomial. For the example polynomial function x^4 - 5x^2 + 4, the x-intercepts are (1,0), (-1,0), (2,0), (-2,0) and the y-intercept is (0,4).
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The document discusses solving quadratic equations by finding the roots or solutions of the equation. It explains that a quadratic equation is of the form ax^2 + bx + c = 0, where a ≠ 0. The roots are the values of x that make the equation equal to 0. To solve the equation, it is set equal to 0 and the square root property, that if x^2 = k then x = ±√k, is applied to find the two roots of the quadratic equation. Several examples are shown step-by-step to demonstrate solving quadratic equations to find their two roots.
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This document discusses factorizing quadratic expressions. It provides examples of expanding and factorizing expressions of the form (x + a)(x + b). A pattern is observed where the constant term is the product of a and b, and the coefficient of x is the sum of a and b. Students are asked to factorize additional quadratic expressions using this pattern.
Solving quadratic equation using completing the square
1. The document provides steps for solving quadratic equations using completing the square. It involves isolating terms with x, finding the square of half the coefficient of x, completing the square of the left side of the equation, applying the square root property, and solving for the roots.
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ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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A Strategic Approach: GenAI in Education
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Your Skill Boost Masterclass: Strategies for Effective Upskilling
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Natural birth techniques are various type such as/ water birth , alexender method, hypnosis, bradley method, lamaze method etcHindi varnamala | hindi alphabet PPT.pdf
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
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Pollock and Snow "DEIA in the Scholarly Landscape, Session One: Setting Expec...
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Any substance (other than food) that is used to prevent, diagnose, treat, or relieve symptoms of a
disease or abnormal condition
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- 1. x + 2y = 10 2x + 3y = 18 Simultaneous Equations
- 2. x + 2y = 10 2x + 3y = 18 x= (18x2)-(10x3) Simultaneous Equations
- 3. x + 2y = 10 2x + 3y = 18 x= (18x2)-(10x3) (2x2)-(3x1) Simultaneous Equations
- 4. x + 2y = 10 2x + 3y = 18 x= (18x2)-(10x3) = 36-30 (2x2)-(3x1) 4-3 Simultaneous Equations
- 5. x + 2y = 10 2x + 3y = 18 x= (18x2)-(10x3) = 36-30 = 6 (2x2)-(3x1) 4-3 1 Simultaneous Equations
- 6. x + 2y = 10 2x + 3y = 18 x= (18x2)-(10x3) = 36-30 = 6 = 6 (2x2)-(3x1) 4-3 1 Simultaneous Equations
- 7. x + 2y = 10 2x + 3y = 18 x= (18x2)-(10x3) = 36-30 = 6 = 6 (2x2)-(3x1) 4-3 1 Putting x=6 in above equation we get, y=2 Simultaneous Equations
- 8. x + 3y = 11 2x + 3y = 13 Simultaneous Equations
- 9. x + 3y = 11 2x + 3y = 13 x= (13x3)-(11x3) Simultaneous Equations
- 10. x + 3y = 11 2x + 3y = 13 x= (13x3)-(11x3) (3x2)-(3x1) Simultaneous Equations
- 11. x + 3y = 11 2x + 3y = 13 x= (13x3)-(11x3) = 39-33 (3x2)-(3x1) 6-3 Simultaneous Equations
- 12. x + 3y = 11 2x + 3y = 13 x= (13x3)-(11x3) = 39-33 = 6 (3x2)-(3x1) 6-3 3 Simultaneous Equations
- 13. x + 3y = 11 2x + 3y = 13 x= (13x3)-(11x3) = 39-33 = 6 = 2 (3x2)-(3x1) 6-3 3 Simultaneous Equations
- 14. x + 3y = 11 2x + 3y = 13 x= (13x3)-(11x3) = 39-33 = 6 = 2 (3x2)-(3x1) 6-3 3 Putting x=2 in above equation we get, y=3 Simultaneous Equations
- 15. 4x + 5y = 14 3x + 3y = 9 Simultaneous Equations
- 16. 4x + 5y = 14 3x + 3y = 9 x= (9x5)-(14x3) Simultaneous Equations
- 17. 4x + 5y = 14 3x + 3y = 9 x= (9x5)-(14x3) (3x5)-(3x4) Simultaneous Equations
- 18. 4x + 5y = 14 3x + 3y = 9 x= (9x5)-(14x3) = 45-42 (3x5)-(3x4) 15-12 Simultaneous Equations
- 19. 4x + 5y = 14 3x + 3y = 9 x= (9x5)-(14x3) = 45-42 = 3 (3x5)-(3x4) 15-12 3 Simultaneous Equations
- 20. 4x + 5y = 14 3x + 3y = 9 x= (9x5)-(14x3) = 45-42 = 3 = 1 (3x5)-(3x4) 15-12 3 Simultaneous Equations
- 21. 4x + 5y = 14 3x + 3y = 9 x= (9x5)-(14x3) = 45-42 = 3 = 1 (3x5)-(3x4) 15-12 3 Putting x=1 in above equation we get, y=2 Simultaneous Equations