Section 2.3 Solving Linear Equations

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1 Variable: Defined as A symbol (or letter) that is used to represent an unknown numbers Examples: a, b, c, x, y, z, s, t, m, n, Constant: Defined as A single number Examples: 1, 2, 3, 6, 1, π, e, π, 1.6, 5/8, Coefficient: Defined as A number written next to the variable 3y, 3 is the coefficient of y 5x 2, 5 is the coefficient of x 2 Identify each term of the algebraic expressions below Expressions Variable(s) Coefficient(s) Constant x + 3 x 1 3 4x 7y +5 x, y 4, x 2 5 x m 6 m x 50t 5s x, t, s 11, -50, -5 None m a m, a 1, -1 None 0.5a 3.7c a, c 0.5, -3.7 None v + 8 v -1 8 Algebraic Expression: Defined as A combination of variables and numbers using any of the operations of +,,, and exponents xy + y Examples: x + 1, 4x 7xy 5, x +,, 2 8 x 7 Like Terms: Terms that are constants or terms that contain the same variables raised to the same exponents We can combine and simplify only like terms. Like Terms as constant: 4, 1.54, 374, 0.37, π, Like Terms as variable term: 5t, 20t, 3.4t, πt Note that 3st 2, 4st, 3s 3 t, and 7s 3 t 2 are not like terms since variables are all different Steps for simplifying expressions: Step 1: Identify each like term Step 2: Rearrange the expression by like terms (Note: only one type of like term in expression, then skip this step) Step 3: Combine like terms by the distributive property Step 4: Add (or subtract) the coefficient and keep the common variable expression Cheon-Sig Lee Page 1

2 Simplify Algebraic Expressions: Steps 3x + 5x 3x 7y 5x + 1 2x + 2 7xy 5x + x 2 y 8xy Step 1 One type (Identifying like terms) Three Types Three Types Step 2 (Rearrangement) =3x 5x 7y + 1 = 2x 5x + 2 7xy 8xy + x 2 y Step 3 (Distribution Law) = (3 + 5)x =(3 5)x 7y + 1 =( 2 5)x ( 7 8)xy + x 2 y Step 4 (Simplifying) = 8x = 2x 7y + 1 = 7x xy + x 2 y Steps for Evaluating Expressions Step 1: Simplify a given expression, if possible Step 2: Substitute the values given for any variable Step 3: Evaluate the resulting expression Evaluate Algebraic Expressions bellow for x = 2, y = 1 Steps 3x + 4 3x 7y 5 x + 1 2x + 2 7xy 5x + x 2 y 8xy Step 1 (Simplifying Expression) = 2x 7y + 1 = 7x 15xy + x 2 y + 2 Step 2 (Substituting) =3(2) + 4 = 2(2) 7(-1) + 1 = 7(2) 15(2)(-1) + (2) 2 (-1) + 2 Step 3 (Evaluating) =6 + 4 =10 = =4 Equation: A statement that two algebraic expressions are equal = =14 Solution: Any number that gives a true statement when substituted for the variable Solution Set: The solutions to an equation form Solving Equations Step 1: Simplify the algebraic expression on each side Step 2: Collect all the variable terms on one side and all the constant terms on the other side Step 3: Isolate the variable and solve. Step 4: Check the proposed solution in the original equation Cheon-Sig Lee Page 2

3 Solve the equations: Steps x + 4= 6 3x 4 2x + 2 = 6 x 6 4x 4 7x + 2 = 3x 8 3x 2x 4 + 2= 6 4x 7x = 3x 8 Step 1 x 2= 6 3x 2= 3x 8 x + 4= 6 x 2= 6 3x 2= 3x x +3x Step 2 x= 2 x= 4 2= 6x = 6x x Step 3 1 = = 6x 6 x= 6 1= x Step 4 Optional Optional Optional Optional Types of Linear Equations: Types Number of solutions Other terms Conditional Finite Number of Solutions One solution Identity Infinite Number of Solutions Infinitely many solutions Contradiction No Solution No solution Exercises (Solution 1) 5x + 8x 7x= x 7x = 24 6x= 24 6x 6 = 24 6 x= 4 (Solution 2) 5(2x 1) = 20 5(2x 1) = x 1 = x = 5 2x 2 = 5 2 x = 5 2 Cheon-Sig Lee Page 3

4 (Solution 3) 16x (9x 5) = 40 16x 9x + 5 = 40 7x + 5= x= 35 7x 7 = 35 7 x = 5 (Solution 4) 13(17 x)= 15(6x + 1) ( x)= 15 6x x = 90x x +13x 221= 103x = 103x = 103x 103 2= x (Solution 5) 4(x + 1)= 7(x 1) 7 4 x + 4 1= 7 x + 7( 1) 7 4x + 4= 7x 7 7 4x + 4= 7x 14 4x 4x 4= 3x = 3x 18 3 = 3x 3 6 = x Cheon-Sig Lee Page 4

5 (Solution 6) 37= 2(23 y) + 3(y 4) 37= ( 2)(23) + ( 2)( y) + 3 y + 3( 4) 37= y + 3y 12 37= 2y + 3y = 5y = 5y 95 5 = 5y 5 19= y (Solution 7) Multiply LCD on both sides; LCD = 3 x 3 2 = 7 3 x 2 3= x 6 = x = 15 (Solution 8) LCD of 4, 5, and 20 is 20 Multiply LCD on both sides; LCD = 20 x 4 + x 5 = x x 5 = x + 4 x = 9 9x= 9 9x 9 = 9 9 x = 1 Cheon-Sig Lee Page 5

6 (Solution 9) LCD of 4 and 9 is 36 Multiply LCD on both sides; LCD = y = y y = 4y 5 4 9y + 20= 4y 20 4y 4y 5y + 20= y = 40 5y 5 = 40 5 y = 8 (Solution 10) LCD of 2 and 3 is 6 Multiply LCD on both sides; LCD = 6 x = x x 1 x 1 6 2= (x 1) 12 = 2(x 1) 3 x + 3 ( 1) 12 = 2 x + 2( 1) 3x 3 12 = 2 x 2 3x 15 = 2x 2 2x 2x x 15= x = 13 (Solution 10) 3.6x= 2.4x x 2.4x 1.2x= x 1.2 = x = 7 Cheon-Sig Lee Page 6