Math 017 Materials With Exercises

2Equations x 3 0 and 4 3 2x 5 are not linear equations. No algebraic operation would „remove‟ x3or x from the equation, and hence each of the equations can never be written to match the desiredform ax b 0.Example 11.8 Determine if the following equations are linear in one variable. If so, express them inthe form ax b 0, a 0 . Determine the values of a and b in your representation.x2a) 1 0 b) 7x 2 3xc) 2x(x 3) 2xd) 3x2 1 04Solution:1a) It is a linear equation since it can be rewritten it in the form x ( 1) 0;41 1 a 0 , b 1. Notice, that this representation is not unique. If we multiply4 4 xboth sides of the equation 1 0 by 4, we get x 4 0 . This can be rewritten as41x ( 4) 0 , and then a 1, b 4. Both answers would be correct. There are otherrepresentations possible.b) Subtract 3 x from both sides 7x 2 3x 0 . Collect like terms 4x 2 0.This matches the form ax b 0, with a 4 ( 4 0), b 2. Thus 7x 2 3xis alinear equation in one variable.222c) Remove parentheses 2x 6x 2x, subtract 2x from both sides222x 6x 2x 0, and simplify to obtain 6x 0. We rewrite 6x 0 as 6x 0 0(adding 0 does not change the value of the expression, but allows us to match the form).2Thus a 6 ( 6 0), b 0 in our representation, and 2x(x 3) 2xis a linearequation in one variable.2d) Since no operation can remove x from the equation, it is not a linearequation.Factoring with the use of the Difference of Squares FormulaConsider the expression ( a b)(a b). When we remove parentheses and simplify, we get22 2 2( a b)(a b) a ab ba b a bIt is called the Difference of Squares Formula.The Difference ofSquaresFormula2a b2 ( a b)(a b)Let us consider the expression 2 2x 9 . Since 9 3 , we can rewrite x 2 9 in its equivalent form,2 22 2x 3 . The expression x 3 matches the left-hand side of the Difference of Squares Formula with109