Transcript Ch 4
Chapter 4 Equations 4.1 Equations and Their Solutions Differentiate between an expression and an equation Check a given number to see if it is a solution for a given equation Example Expression 2x + 5 Read : ‘ two x plus five ‘ Equation 2x + 5 = 4 ‘two x plus five is equal to 4 Definition Solve: To find the solution or solutions to an equation Solution: A number that makes an equation true when it replaces the variable in the equation Procedure To check to see if a value is a solution to an equation 1. 2. Replace the variable(s) with the value Simplify both sides of the equation as needed. If the resulting equation is true, then the value is a solution Examples Example 1 Is 2 a solution of for 2x – 7 = 5 ? 2(2) – 7 ?= 5 ( Replace x with 2 ) 4 – 7 ?= 5 (Simplify ) - 3 = 5 ( False) So the resulting equation is not true, 2 is not a solution for 2x – 7 = 5 Example 2 Is -7 is a solution for 3x + 5 = 2(x – 1) ? 3( - 7) + 5 ?= 2((-7) – 1) ( Replace x by – 7) -21 + 5 ?= 2(- 8) ( Simplify ) - 16 = - 16 ( True ) So the resulting equation is true, - 7 is a solution for 3x + 5 = 2(x – 1) 4.2 The addition/Subtraction Principle of Equality Determine whether a given equation is linear Definition Linear equation : An equation that is made of polynomials or monomials that are at most degree 1. Examples 2x – 3y = 7 ( Linear ) y = x2 ( Non linear ) Solve linear equations in one variable using the addition /subtraction principle of equality. Rule 1. The addition /subtraction principle of equality 2. The same amount can be added to or subtract from both sides of an equation without affecting its solution(s). Procedure To use the addition/subtraction principle of equality to clear a term, add the additive inverse of that term to both sides of the equation( that is, add or subtract appropriately so that the term you want to clear becomes 0) Example Solve and Check 3( x + 2) = 4 + 2(x – 1) 3x + 6 = 4 + 2x – 2 ( Distribute to clear parenthesis ) 3x + 6 = 2x + 2 - 2x - 2x x +6=0+2 ( Combine 3x and – 2x to get x) -6 -6 ( Add – 6 both sides to isolate x) x =-4 Check 3( x + 2) = 4 + 2(x – 1) 3( - 4 + 2) ?= 4 + 2( - 4 – 1) ( Replace x by -4) 3 ( - 2) ? = 4 + 2 ( - 5) ( Simplify) - 6 ?= 4 - 10 -6=-6 The equation is true, so – 4 is the solution. Solve equations with variables on both sides of the equal sign To solve equations: 1. a) Simplify both sides of the equation as needed. b) Distribute to clear parentheses c) Combine like terms. 2. Use the addition /subtraction principle so that all variable terms are on side of the equation and all constants are on the other side. Then combine like terms. 4.3 The Multiplication/Division Principle of Equality Solve equations using the multiplication/division principle of equality Rule The multiplication/Division Principle of equality We can multiply or divide both sides of an equation by the same nonzero amount without affecting its solution(s) Procedure To use the multiplication/division principle of equality to clear a coefficient, divide both sides by that coefficient. Solve equations using both the addition/subtraction and the multiplication/division principles of equality To solve equations 1. a) b) 2) 3) Simplify both sides of the equation as needed. Distribute to clear parentheses Combine like terms Use the addition /subtraction principle of equality so that all variable terms are on one side of the equation and all constants are on the other side. (Clear the variable term with the lesser coefficient.) Then combine like terms. Use the multiplication/division principle of equality to clear the remaining coefficient. Solve application problems Area of a parallelogram: A = bh Volume of a box: V = lwh Force: F = ma Distance: d = rt Voltage: V = ir Perimeter of a rectangle: P = 2l + 2w Surface area of a box: SA = 2lw+ 2lh + 2wh 4.4 Translating word sentences to Equations Key words for an equal sign Is equal to Produces Is the same as Yields Is Results in