GLOSSARY equality simultaneous equations (Br.) system of equations (Am.) conditional equation solutions (of several simultaneous equation) degree of an equation to verify a solution to check a solution to solve simultanous equations to rearrange a formula to rearrange an equation into the form… to change the subject of a formula to substitute a value into an equation to isolate a variable to plug in a number for a letter quadratic equation linear equation radical equations a substitution story/word problem to represent some quantity with a variable to sketch a figure the elimination method the substitution method dependent system of linear equations independent system of linear equations inconsistent system of linear equations consistent system of linear equations LESSON 5 Simultaneous equations Simultaneous equations A set of simultaneous equation consist of two or more equations. They usually have more than one variables or unknowns. A set of values for the variables is a solution of the simultaneous equations when all the equations become true if the values replace the variables. x3 x 3 x3 x2 9 Examples: is a solution, is a solution, is not a solution y7 y 13 y 8 x y 10 Verifying solutions A solution of a set of simultaneous equations can be verified by substituting the values for the variables. Classifying sets of simultaneous equations According to the number of equations and variables (2, 3, 4, ... equations; 1, 2, 3, 4, ... variables) According to the type and degree of the equations (polynomic, linear, non-linear, rational, irrational...) According to the number of solutions Solving simultaneous equations means finding the answers to several equations at the same time. The first type of simultaneous equations we are going to solve is a set of two first-degree equations (linear equations) in two variables. That is any set of simultaneous equations equivalent to one like this: ax by c We must find values for “x” and “y” for which both equations are true. a ' x b ' y c ' Solving two linear simultaneous equations with two unknowns Always before applying any method: -Rearrange both equations into the above form. Always after applying any method: -Once you get the value of one of the unknowns, substitute it into any equation to get the other. The substitution method: -Leave one variable alone rearranging the easiest equation. -Plug in the resulting expression for that variable into the other equation. -Solve the equation obtained. 2 x 16 2 y Example: 2 x 3 y 16 2 x 16 y 2 2 x 3 y 16 x8 y 2 x 3x 8 16 2x 3x 24 16 2x 3x 16 24 x 40 x 40 2 40 16 y 2 80 16 y 2 64 y 2 32 y Another method: -Isolate one variable rearranging the first equation. -Isolate the same variable rearranging the second equation. -Form a new equation equalling both expressions and solve it. 7 x 2 y 8 Example: 5x 3 y 1 7 x 8 2 y 5x 1 3 y 8 2y x 7 1 3y x 5 8 2 y 1 3y 7 5 40 10 y 7 21 y 35 35 40 10 y 7 21y 10 y 21y 7 40 11y 33 33 11 y3 y 1 33 5 10 x 5 x2 x The elimination method: -If both coefficients of “x” are opposite numbers (or both coefficients of “y” are) add the equations and solve the resulting equation. -If not, try to reach that situation multiplying one of the equations by a suitable number. -Sometimes you have to change both equations multiplying each one by a different number (the coefficient of the like term in the other equation or else its opposite). x 3 y 21 Example: 2 x 5 y 35 2 x 6 y 42 2 x 5 y 35 0 x 11y 77 11y 77 77 y 11 y 7 x 3 7 21 x 21 21 x 21 21 x0 Homework: 3abc page 135 Class work: Homework: 4bd page 135 4ace page 135 Class work: Homework: 5ac page 135 5b and 7cd page 135 Class work: Classwork: 7ab page 135 which method would fit better to each pair of simultaneous equations in exercise 6 page 135? How to solve “story/word problems” Steps: - Read the problem. This includes identifying all the given information and identifying what you being asked to find. - Represent each of the unknown quantities with a variable. - If applicable, sketch a figure illustrating the situation. - Form an equation that will relate known quantities to the unknown quantities. To do this make use of known formulae. - Solve the simultaneous equations formed in the previous step and write down the answer to all the questions. - It is important to answer all the questions that you were asked. Often you will be asked for several quantities. - Check your answer and make sure that the answer makes sense. Homework: 8 page 135 Class work: Homework: 14 page 136 9 and 26 pages 135 to 137 Class work: Homework: 16 page 136 18 page 136 Class work: Homework: 19 and 29 pages 135 to 136 31 and 32 page 136 Classwork: 1) The admission fee at a small fair is 1.50€ for children and 4.00€ for adults. On a certain day, 2200 people enter the fair and 5050€ is collected. How many children and how many adults attended? 2) The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number. 3) Find the equation of the parabola that passes through the points (–1, 9), (1, 5), and (2, 12). 4) A landscaping company placed two orders with a nursery. The first order was for 13 bushes and 4 trees, and totalled 487€. The second order was for 6 bushes and 2 trees, and totalled 232€. The bill does not list the per-item price. What is the cost of one bush and of one tree? 5) A passenger jet took three hours to fly 1800 miles in the direction of the jetstream. The return trip against the jetstream took four hours. What was the jet's speed in still air and the jetstream's speed? Number of solutions of two simultaneous linear equations An independent system of linear equations is that with a single solution. A dependent system of linear equations is that with a non-finite set of solutions. One equation is a b c really just another copy of the other: coefficients and constant term are proportional a ' b' c ' An inconsistent system of linear equations is that without any solution. That happens when a b c coefficients are proportional but the constant terms are not in the same proportion as coefficients a ' b' c ' Examples: 3x 5 y 4 6 x 10 y 8 3 5 4 6 10 8 dependent system x 3y 9 x 2 y 5 1 3 1 2 independent system 5x y 4 5 x 1 y 5 5 x y 4 5 x y 4 5 1 4 5 1 4 2 x 5 y 11 2x 5 y 3 2 5 11 2 5 3 inconsistent system 5x y 4 10 x 2 y 4 5 1 4 10 2 4 inconsistent system x 3 y 11 2 x 6 y 21 1 3 11 2 6 21 inconsistent system dependent system Simultaneous equations of second degree Substitution method is to be used. y x 1 2 x 2 x 1 5 Example: 2 2 x y 5 x 2 x 2 2x 1 5 2x 2 2x 4 0 x2 x 2 0 x Homework: 45 page 139 1 12 4 1 2 2 1 x 1, y 2 1 1 8 1 3 x 2, y 1 2 2

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# Beginnings of Algebra