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Maths Quest 10 for New South Wales 5.3 Pathway Work Programs Chapter 8 Simultaneous equations Strand: Patterns and algebra Substrands and outcomes: Algebraic techniques PAS4.3 Uses the algebraic symbol system to simplify, expand and factorise simple algebraic expressions Algebraic techniques PAS4.4 Uses algebraic techniques to solve linear equations and simple inequalities Algebraic techniques PAS5.2.2 Solves linear and simple quadratic equations, solves linear inequalities and solves simultaneous equations using graphical and analytical methods Algebraic techniques PAS5.3.2 Solves linear, quadratic and simultaneous equations, solves and graphs inequalities, and rearranges literal equations Coordinate geometry PAS5.2.3 Uses formulae to find midpoint distance and gradient and applies the gradient/intercept form to interpret and graph straight lines Coordinate geometry PAS5.3.3 Uses various standard forms of the equation of a straight line and graphs regions on the number plane Curve sketching and polynomials PAS5.3.6 Uses a variety of techniques to sketch a range of curves and describes the features of curves from the equation Section Are you ready? (page 290) GC tips, Investigations, History of mathematics, Maths Quest challenge, 10 Quick Questions, Code puzzles, Career profiles SkillSHEETS, WorkSHEETS, Interactive games, Test yourself, Topic tests (CD ROM) SkillSHEETs (page 290) 8.1: Solving linear 1 Technology applications (CD ROM) Learning outcomes PAS4.4 solving equations arising equations that arise when finding x- and yintercepts 8.3: Finding the gradient of a line from its equation 8.4: Transposing linear equations to make x or y the subject 8.6: Writing equations from worded statements 8.7: Checking solutions to inequalities Introduction (page 291) Solving simultaneous equations using nonalgebraic methods (page 292) Graphical solution of simultaneous equations (page 294) WE 1, 2, 3 Ex 8A Graphical solution of simultaneous equations (page 298) Investigation: Solving simultaneous equations using tables of values (page 292) Investigation: Solving simultaneous equations using guess, check and improve (page 293) GC tip – Casio: Solving simultaneous equations graphically (page 297) Investigation: How many cockatoos and kangaroos? (page 300) SkillSHEET 8.1: Solving linear equations that arise when finding xand y- intercepts (page 299) SkillSHEET 8.2: Graphing linear equations using the x- and y-intercept method (page 299) SkillSHEET 8.3: Finding the gradient of a line from its equation (page 300) Game time 001 (page 300) 2 GC tip – TI: Solving simultaneous equations graphically (page 297) Mathcad: Simultaneous equations – graphical (page 298) Excel: Simultaneous equations – graphical (page 298) from substitution into formulae checking solutions by substituting PAS5.2.3 recognising equations of the form y mx b as representing straight lines and interpreting the x-coefficient (m) as the gradient PAS5.3.2 changing the subject of a formula PAS4.3 translating from everyday language to algebraic language PAS5.2.2 solving simultaneous equations using nonalgebraic methods, such as ‘guess and check’, setting up tables of values or looking for patterns solving linear simultaneous equations by finding the point of intersection of their graphs using graphics calculators and spreadsheet software to plot pairs of lines and Algebraic solutions of simultaneous equations – substitution method (page 301) WE 4, 5, 6 Ex 8B Algebraic solutions of simultaneous equations – substitution method (page 304) Algebraic solutions of simultaneous equations – elimination method (page 306) WE 7, 8 Ex 8C Algebraic solutions of simultaneous equations – elimination method I (page 307) Maths Quest challenge Q1-5 (page 304) 10 Quick Questions 1 (page 305) Further use of the elimination method (page 308) WE 9, 10 Ex 8D Algebraic solutions of simultaneous equations – elimination method 2 (page 310) GC tip – Casio: Solving simultaneous equations using the equations function (page 309) Maths Quest challenge Q1-2 (page 310) Code puzzle (page 311) Investigation: Cramer’s rule for simultaneous equations (page 312) Investigation: Simultaneous equations in 3 unknowns SkillSHEET 8.4: Mathcad: Simultaneous Transposing linear equations – substitution equations to make x or y (page 304) the subject (page 304) Excel: Simultaneous Game time 002 (page 304) equations – substitution WorkSHEET 8.1 (page 304) (page 304) reading off the point of intersection (Applying strategies) PAS5.2.2 solving simple linear simultaneous equations using an analytical method SkillSHEET 8.5: Addition and subtraction of simultaneous equations (page 307) Mathcad: Simultaneous equations – elimination (page 308) Excel: Simultaneous equations ax + by = c form (page 308) PAS5.2.2 solving simple linear simultaneous equations using an analytical method WorkSHEET 8.2 (page 310) GC tip – TI: Solving simultaneous equations using the equations function (page 309) Excel: Simultaneous equations – elimination (page 310) PAS5.2.2 solving simple linear simultaneous equations using an analytical method solving linear simultaneous equations resulting from problems and interpreting the results (Applying strategies, Communicating) 3 Problem solving using simultaneous equations (page 313) WE 11, 12 Ex 8E Problem solving using simultaneous equations (page 315) (page 312) Investigation: Concert hall seating (page 317) 10 Quick Questions 2 (page 318) Maths Quest challenge Q1-3 (page 318) SkillSHEET 8.6: Writing equations from worded statements (page 315) Solving a quadratic equation and a linear equation simultaneously (page 319) WE 13, 14, 15 Ex 8F Solving a quadratic equation and a linear equation simultaneously (page 322) Excel: Simultaneous equations (page 315) GC program TI: Simultaneous equations (page 315) Mathcad: simultaneous equations – quadratic and linear (page 322) Excel: simultaneous equations – quadratic and linear (page 322) 4 PAS5.2.2 generating simultaneous equations from simple word problems solving linear simultaneous equations resulting from problems and interpreting the results (Applying strategies, Communicating) PAS5.3.2 using analytical methods to solve a variety of simultaneous equations, including those that involve a first degree equation and a second degree equation explaining why a particular value could not be a solution to an equation (Applying strategies, Reasoning, Communicating) graphing simultaneous equations to find solutions and comparing this method with analytical methods (Applying strategies, Reflecting, Communicating) PAS5.3.6 Solving simultaneous inequalities (page 323) WE 16 Ex 8G Solving simultaneous inequalities (page 326) SkillSHEET 8.7: Checking solutions to inequalities (page 326) SkillSHEET 8.8: Checking whether a given point makes the inequality a true statement (page 326) SkillSHEET 8.9: Shading half planes to demonstrate linear inequalities (page 326) WorkSHEET 8.3 (page 329) 5 GrafEq (page 325) determining the points of intersection of a line with a parabola graphically and algebraically PAS5.3.2 using <, >, , , to generate linear inequalities from problems PAS5.3.3 graphing inequalities of the form y < a, y > a, y a, y a, x > a, x a and x a on the number plane graphing inequalities such as y x on the number plane by considering the position of the boundary of the region as the limiting case checking whether a particular point lies in a given region specified by a linear inequality graphing regions such as that specified by x+y<7 2x 3y 5 describing the region from a graph by identifying the boundary lines and determining the appropriate inequalities for describing the enclosed region (Applying strategies, Communicating) Summary (page 330) Chapter review (page 331) ‘Test yourself’ multiple choice questions Topic tests (2) 6