"Note Sheet #3"
Note Sheet #3 Solving Linear Systems Algebraically Rules of exponents and Integer exponents Vocabulary Consistent – System has a solution (the line intersect) Inconsistent – System does not have a solution (ie parallel lines) Dependent - infinite number of solutions (the equations graph the same line!) Algebraic Methods The Algebraic Methods are designed to find the coordinates of the points of intersection that you would find if you graphed the two lines (the solution to the system of equations). Both algebraic methods find ONLY one half of the solution. To find the "partner" in the ordered pair you will need to evaluate one of the equations with using the value you found. You should check your work by evaluating the other equation using the ordered pair to verify your work. Substitution Method: You must solve one of the equations for one of the variables. (This is the preferred method is used when one of the coefficients is a ONE to avoid fractions or is already solved for one of the variables). Then you "substitute" the information from that equation into the second equation. Example: x = 3y - 1 (EQ. 1) 2x + 5y = 7 (EQ. 2) 2(3y - 1) + 5y = 7 (Combo of EQ. 1 & 2) Elimination Method: Both equations must be written in standard form. ( Ax + By = C) The object is to add to two equations together and in the process one of the variables (letters) is eliminated. Most of the time we have to multiply everything on both sides of the equal sign by a number so that they will "cancel out" Example 1: 3x + 2y = 5 Example 2: -3x + 2y = 5 (EQ. 1) 5x - 2y = 11 4x - 3y = -9 (EQ. 2) 8x = 16 x=2 3(-3x + 2y) = (5)3 (EQ 1) 2(4x - 3y) = (-9)2 (EQ 2) -9x + 6y = 15 (EQ. 1) 8x - 6y = -18 (EQ. 2) x = -3 (Combo) Important Neither Algebraic Method you the solution. It gives you ½ of the solution. You need to take the coordinate and find the other ½ of the ordered pair.by “plugging it in” to one of the original equations. Always check your final solution in in the “other equation”. Exponential Notation - a "shorthand" way of writing a multiplication problem. Exponent only effects the number its next too. If it is to effect more than just the number it has to be "bundled together" (bundle)exponent Definition Exponents Positive exponent negative exponent zero exponents 1 x4 = (x)(x)(x)(x) x −3 = 3 x0 = 1, x ≠ 0 x It is helplful to recognize the pattern of a negative exponent in the denominator of a fraction. 2 2 x3 3x −2 3 y 3 3x 2 y −1 3x 2 w3 = = 2 x3 = 2 = x −3 1 5 y −3 5 x w−3 z 5 yz 5 Your notes: Product, Quotient and Power “rules” are just recognizing and using patters that occur when we apply the definitions of exponents. x 5 x 7 = x12 Product Rule x8 x5 x5 = x3 x 8 = 3 x 1 ( aka x −3 ) Quotient Rule (x ) 2 4 = x8 Power Rule Your Notes: