03+Solving+Equations

Section 3.1- 3.5 : Solving Equations:

**I. One Step Equations:**

3:1 – A. Translate Sentences into Equations Examples: a). 2 times a number //x// is equal to 4 times the sum of b and c 1). 2(//x)// = 4 x (b + c) //- Answer: the equation is 2x = 4(b + c)// B. Translate Equations into Sentences a). 5a + 5 = 15 1). - //Answer: 5 times a plus 5 equals 15// b). b + c = k3 1). //– Answer: The sum of b and c = k cubed//

3:2 – A. Solving Equations by using addition and subtraction a). // Solve the Equation : x – 24 = 48// 1). First add 24 to both sides 2). By adding 24 it cancels out on the left 3). Now the equation becomes //x = 72// b). //Solve 142 + h = 97// 1). First subtract 142 to both sides 2). By subtracting 142 it cancels out on the left 3). Now the equation becomes //h = - 45//

3:3 – A. Solving Equations by using multiplication and division a). // Solve the Equation: // //t_ __= 7___// // 30  10 //  1). Multiply 30 to both sides 2). //– Answer:// //t = 21// b). // Solve the Equation : 13d = 195// 1). Divide 13 from both sides 2). //– Answer: d = 15 // c). // Solve the Equation : -3x = 12// 1). Divide – 3x from both sides 2). //– Answer x = - 4//

II. Multi-Step Equations: Section 3.4 pages 142 - 148 **
 * To solve an equation with more than one operation, you undo operations by working backwards.


 * Solve Using: Addition and Division **
 * Example 1:**

5//x// - 3 = 12 +3 +3 > add 3 to both sides of the equal sign 5//x// = 15 > the 3 in the original equation cancels out and you are left with the 5 // x // and the 15 /5 /5 > divide both sides by 5 //x// = 3 > when you divide 5 by itself it cancels out leaving you only to divide 15 by 5, giving you your answer, 3 // y///8 + 21 = 14 ** -21 -21 > subtract 21 from both sides of the equal sign //y///8 = -7 > the 21 then cancels out and when you subtract 21 from 14, you get a negative number 8(//y///8) = 8(7) > now, multiply everything by the denominator, 8 //y// = -56 > you then get a negative answer because a negative times a positive is always negative
 * This type of equation also applies with fractions
 * Solve Using: Subtraction and Multiplication **
 * Example 2:

Example 3:**
 * Solve Using: Multiplication and Addition

//x// - 15/9= -6 > the original equation //x// - 15 = -54 > you then get rid of the fractions and are then left with what you see here +15 +15 > now add both sides of the equal sign by 15 //x// = -39 > then you are able to solve for // x // **
 * 9(//x// - 15/9) = 9(-6) >multiply both sides of the equal sign by 9

Now that you have seen examples of Multi-Step Equations, let's try and write it out by reading a word problem

Two-thirds of a number minus six is negative ten 2/3x - 6 = -10 of - multiplication minus - subtraction is - equals
 * Example 4:**

III. ** Special Equations ( No Solution and All Real Numbers) ** //Examples:// A. //No solution// 1). 2m+5=5(m-7)-3m (original equation) 2). 2m+5=5m-35-3m 3). 2m+5=2m-35 4). 2m+5-2m=2m-35-2m 5). – Answer - 5=-35 which becomes No Solution because – 5 doesn’t equal -35
 * // - - Section 3:5 //**

B. //No solution// 1). 4(f-2)=4f+9 (original equation) 2).4f-8=4f+9 3). -4f -4f 4). -8=9 which becomes no solution because -8 doesn’t equal 9

//Examples:// A. //All Real Numbers// 1). 3(r+1)-5=3r-2 (original equation) 2). 3r+3-5=3r-2 3). 3r-2=3r-2, Both the sides of the equation equal each other which makes the equation become All Real Numbers

B. //All Real Numbers// 1). 3(1+d)-5=3d-2 (original equation) 2). 3+3d-5=3d-2 3). 3d-2=3d-2, Both the sides of the equation equal each other which makes the equation become All Real Numbers