# SS 2 Mathematics(1st Term) ## SIMULTANEOUS EQUATIONS 3

SIMULTANEOUS EQUATIONS See also SIMULTANEOUS EQUATIONS SIMULTANEOUS EQUATIONS QUADRATIC EQUATIONS GEOMETRIC PROGRESSION ARITHMETIC PROGRESSION (A. P) ## SIMULTANEOUS EQUATIONS 2

SIMULTANEOUS EQUATIONS CONTENT Solving Simultaneous Equations Involving One linear and One quadratic. Solving Simultaneous Equations Using Graphical Method SIMULTANEOUS EQUATIONS INVOLVING ONE LINEAR AND ONE QUADRATIC One of the equations is in linear form while the other is in quadratic form. Note: One linear, one quadratic is only possible analytically using substitution method.   Examples: Solve simultaneously for x and y (i.e. the points of their intersection) 3x + y = 10 & 2×2 +y2 = 19 Solution 3x + y = 10 ———– eq 1 2×2 + y2 = 19 ——— eq 2   Make y the subject in eq 1 (linear equation) y = 10 – 3x ———- eq 3 Substitute eq 3 into eq 2 2×2 + (10-3x) 2   = 19 2×2+ (10 – 3x) (10 – 3x) = 19 2×2 + 100 – 30x – 30x + 9×2 = 19 2×2 + 9×2 – 30x –… Read More »SIMULTANEOUS EQUATIONS 2 ## SIMULTANEOUS EQUATIONS

SIMULTANEOUS EQUATIONS CONTENT Solving Simultaneous Equations Using Elimination and Substitution Method Solving Equations Involving Fractions. Word problems. SIMULTANEOUS LINEAR EQUATIONS Methods of solving Simultaneous equation Elimination method Substitution method iii.         Graphical method   ELIMINATION METHOD One of the unknowns with the same coefficient in the two equations is eliminated by subtracting or adding the two equations. Then the answer of the first unknown is substituted into either of the equations to get the second unknown.   Example Solve for x and y in the equations 2x + 5y   = 1  and 3x – 2y   =   30 Solution To eliminate x multiply equation 1 by 3 and equation 2 by 2 2x + 5y = 1      ……….                           eqn 1 (x (3) 3x – 2y = 30 …………                eqn 2 (x (2) Resulting into, 6x + 15 y = 3   ……….               eqn 3 6x – 4y = 60   ………..  eqn 4 Subtract… Read More »SIMULTANEOUS EQUATIONS QUADRATIC EQUATIONS CONTENT Construction of Quadratic Equations from Sum and Product of Roots. Word Problem Leading to Quadratic Equations.   CONSTRUCTION OF QUADRATIC EQUATIONS FROM SUM AND PRODUCT OF ROOTS We can find the sum and product of the roots directly from the coefficient in the equation. It is usual to call the roots of the equation α and β  If the equation ax2 +bx + C = 0   …………….  I has the roots α and β then it is equivalent to the equation (x – α  )( x – β )  = 0 x2 – βx –  βx + αβ  = 0 ………… 2 Divide equation (i)by the coefficient  of x2  ax2+ bx + C     = 0  …………   3 aaa Comparing equations (2) and (3) x2 +  b x  +  C     = 0 aa x2  –  ( α +β)x  + αβ    = 0 then α+ β= -b a and… Read More »QUADRATIC EQUATIONS ## GEOMETRIC PROGRESSION

GEOMETRIC PROGRESSION CONTENT Definition of Geometric Progression Denotations of Geometric progression The nth term of a G. P. The sum of Geometric series Sum of G. P. to infinity Geometric mean Definition of G. P The sequence 5, 10, 20, 40 has a first term of 5 and the common ratio Between the term is 2 e.g. (10/5 or 40/2o = 2). A sequence in which the terms either increase or decrease in a common ratio is called a Geometric Progression (G. P) P: a, ar,     ar2,        ar3 ………………   Denotations in G. P a    = 1st term r    = common ratio Un = nth term Sn  = sum   The nth term of a G. P The nth term = Un Un = arn-1 1st term   = a 2nd term = a x r =ar 3rd term = a x r x r = ar2 4th term = a… Read More »GEOMETRIC PROGRESSION ## ARITHMETIC PROGRESSION (A. P)

ARITHMETIC PROGRESSION (A. P) CONTENT Sequence Definition of Arithmetic Progression Denotations in Arithmetic progression Deriving formulae for the term of A. P. Sum of an arithmetic series Find the next two terms in each of the following sets of number and in each case state the rule which gives the term. (a)        1, 5, 9, 13, 17, 21, 25(any term +4 = next term) (b)        2, 6, 18, 54, 162, 486, 1458 (any term x 3 = next term) (c)        1, 9, 25, 49, 81, 121, 169, (sequence of consecutive odd no) (d)        10, 9, 7, 4, 0, -5, -11, –18, -26, (starting from 10, subtract 1, 2, 3 from immediate no).   In each of the examples below, there is a rule which will give more terms in the list. A list like this is called a SEQUENCE in many cases; it can simply matter if a general term… Read More »ARITHMETIC PROGRESSION (A. P) ## PERCENTAGE ERROR

PERCENTAGE ERROR See also LOGARITHM OF NUMBERS MENSURATION Symmetric Properties of Roots Sum & Product of Roots of a Quadratic Equation Quadratic Formula ## LOGARITHM OF NUMBERS

REVISION OF LOGARITHM OF NUMBERS GREATER THAN ONE AND LOGARITHM OF NUMBERS LESS THAN ONE See also MENSURATION Symmetric Properties of Roots Sum & Product of Roots of a Quadratic Equation Quadratic Formula Completing the square method

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