Solving One-Step Equations Section 2-1

Goals Goal Rubric To solve one-step equations in one variable. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Equivalent Equations Addition Property of Equality Subtraction Property of Equality Isolate Inverse Operations Multiplication Property of Equality Division Property of Equality

Linear Equations in One Variable A linear equation in one variable is an equation that can be written in the form ax + b = c where a, b, and c are real numbers and a  0. 3x + 5 = 25 The expressions are called the sides of the equation.

Solutions The solution of a linear equation is the value or values of the variable that make the equation a true statement. The set of all solutions of an equation is called the solution set. The solution satisfies the equation. Example: Determine if x = – 1 is a solution to the equation. – 3(x – 3) = – 4x + 3 – 5x – 3[(– 1) – 3] = – 4(– 1) + 3 – 5(– 1) – 3(– 4) = 4 + 3 + 5 12 = 12 True. x = – 1 is a solution

Equivalent Equations Linear equations are solved by writing a series of steps that result in the equation x = a number One method for solving equations is to write a series of equivalent equations. Two or more equations that have precisely the same solutions are called equivalent equations. 3 + 5 = 8 1 + 7 = 2 + 6

Equivalent Equations Two equations are said to be equivalent if every solution of either one is also a solution of the other. The two equations shown above are equivalent, because the number 3 will satisfy both equations and 3 is the only number that will satisfy either equation.

Inverse Operations To find solutions, perform inverse operations until you have isolated the variable. A variable is isolated when it appears by itself on one side of an equation, and not at all on the other side. Isolated Variable x = a number Inverse Operations Add x. Subtract x. Multiply by x. Divide by x. An equation is like a balanced scale. To keep the balance, you must perform the same inverse operation on both sides.

Inverse Operations Inverse Operations Operation Inverse Operation Isolate a variable by using inverse operations which "undo" operations on the variable. An equation is like a balanced scale. To keep the balance, perform the same operation on both sides. Inverse Operations Operation Inverse Operation Addition Subtraction Subtraction Addition

Inverse Operations Inverse Operations Operation Inverse Operation Solving an equation that contains multiplication or division is similar to solving an equation that contains addition or subtraction. Use inverse operations to undo the operations on the variable. Inverse Operations Operation Inverse Operation Multiplication Division Division Multiplication

Addition Property Of Equality The Addition Property of Equality states that for real numbers a, b, and c, if a = b, then a + c = b + c. We need to find the value of y. y - 6 = 11 y - 6 (+6) = 11 (+6) Adding (+6) to both sides of the equation will maintain the balance of the equation. y = 17 Left side Right side +6 Solution to the equation.

Using the Addition Property of Equality Example: Solve the linear equation x  9 = 22. Step 1: Isolate the variable x on the left side of the equation. x  9 + 9 = 22 + 9 Add 9 to both sides of the equation. Step 2: Simplify the left and right sides of the equation. x = 31 Apply the Additive Inverse Property. Step 3: Check to verify the solution. x  9 = 22 31  9 = 22 22 = 22 

3 = 3 3 + 2 = 3 + 2 5 = 5 a = b a + c = b + c Properties of Equality WORDS Addition Property of Equality You can add the same number to both sides of an equation, and the statement will still be true. NUMBERS 3 = 3 3 + 2 = 3 + 2 5 = 5 ALGEBRA a = b a + c = b + c

7 = 7 7 – 5 = 7 – 5 2 = 2 a = b a – c = b – c Properties of Equality WORDS Subtraction Property of Equality You can subtract the same number from both sides of an equation, and the statement will still be true. NUMBERS 7 = 7 7 – 5 = 7 – 5 2 = 2 ALGEBRA a = b a – c = b – c

Writing Math Solution sets are written in set notation using braces, { }. Solutions may be given in set notation, or they may be given in the form x = 14.

Example Solve the equation. y – 8 = 24 y = 32 Check y – 8 = 24 + 8 + 8 Since 8 is subtracted from y, add 8 to both sides to undo the subtraction. y = 32 The solution set is {32}. Check y – 8 = 24 To check your solution, substitute 32 for y in the original equation. 32 – 8 24  24 24

Example Solve the equation. 4.2 = t + 1.8 –1.8 –1.8 2.4 = t Since 1.8 is added to t, subtract 1.8 from both sides to undo the addition. –1.8 –1.8 2.4 = t The solution set is {2.4}. Check 4.2 = t + 1.8 4.2 2.4 + 1.8 To check your solution, substitute 2.4 for t in the original equation.  4.2 4.2

Your Turn: Solve the equation. Check your answer. n – 3.2 = 5.6 Since 3.2 is subtracted from n, add 3.2 to both sides to undo the subtraction. + 3.2 + 3.2 n = 8.8 The solution set is {8.8}. Check n – 3.2 = 5.6 To check your solution, substitute 8.8 for n in the original equation. 8.8 – 3.2 5.6 5.6 5.6 

Your Turn: Solve the equation. Check your answer. –6 = k – 6 + 6 + 6 Since 6 is subtracted from k, add 6 to both sides to undo the subtraction. + 6 + 6 0 = k The solution set is {0}. Check –6 = k – 6 To check your solution, substitute 0 for k in the original equation. –6 0 – 6 –6 –6 

Your Turn: Solve the equation. Check your answer. 6 + t = 14 – 6 – 6 – 6 – 6 Since 6 is added to t, subtract 6 from both sides to undo the addition. t = 8 The solution set is {8}. Check 6 + t = 14 To check your solution, substitute 8 for t in the original equation. 6 + 8 14  14 14

Multiplication Property of Equality The Multiplication Property of Equality states that for real numbers a, b, and c, where c  0, if a = b, then ac = bc. We need to find the value of x. Multiplying both sides of the equation by will maintain the balance of the equation. x = 28 Right side Left side × 7 Solution to the equation.

Using the Multiplication Property of Equality Example: Solve the linear equation 3x = 81 Step 1: Get the coefficient of the variable x to be 1. (3x) = (81) Multiply each side of the equation by Step 2: Simplify the left and right sides of the equation. x = 27 Apply the Multiplicative Inverse Property. Step 3: Check to verify the solution. 3x = 81 3(27) = 81 81 = 81 

6 = 6 6(3) = 6(3) 18 = 18 a = b ac = bc Properties of Equality WORDS Multiplication Property of Equality You can multiply both sides of an equation by the same number, and the statement will still be true. NUMBERS 6 = 6 6(3) = 6(3) 18 = 18 ALGEBRA a = b ac = bc

Properties of Equality Division Property of Equality You can divide both sides of an equation by the same nonzero number, and the statement will still be true. WORDS a = b (c ≠ 0) 8 = 8 2 = 2 ALGEBRA NUMBERS 8 4 = a c

Example: Solve the equation. Check your answer. –24 = j Check  –8 –8 Since j is divided by 3, multiply from both sides by 3 to undo the division. –24 = j The solution set is {–24}. Check To check your solution, substitute –24 for j in the original equation.  –8 –8

Example: Solve the equation. Check your answer. –4.8 = –6v 0.8 = v Since v is multiplied by –6, divide both sides by –6 to undo the multiplication. 0.8 = v The solution set is {0.8}. Check –4.8 = –6v To check your solution, substitute 0.8 for v in the original equation. –4.8 –6(0.8) –4.8 –4.8 

Your Turn: Solve each equation. Check your answer. p = 50 Check 10 10 Since p is divided by 5, multiply both sides by 5 to undo the division. p = 50 The solution set is {50}. Check To check your solution, substitute 50 for p in the original equation. 10 10 

Your Turn: Solve each equation. Check your answer. y = –20 Check Since y is multiplied by 0.5, divide both sides by 0.5 to undo the multiplication. y = –20 The solution set is {–20}. Check 0.5y = –10 To check your solution, substitute –20 for y in the original equation. 0.5(–20) –10 –10 –10 

Your Turn: Solve each equation. Check your answer. c = 56 Check 7 7  Since c is divided by 8, multiply both sides by 8 to undo the division. c = 56 The solution set is {56}. Check To check your solution, substitute 56 for c in the original equation. 7 7 

Solving Equations When solving equations, you will sometimes find it easier to add an opposite to both sides instead of subtracting or to multiply by a reciprocal instead of dividing. This is often true when an equation contains negative numbers or fractions.

Example: Solve each equation. The reciprocal of is . Since w is multiplied by multiply both sides by . The solution set is {–24}.

{ } Example: Solve each equation. Since p is added to , add to both sides to undo the subtraction. The solution set is . { }

Your Turn: Solve the equation. Check your answer. –2.3 + m = 7 Since –2.3 is added to m, add 2.3 to both sides. +2.3 + 2.3 m = 9.3 The solution set is {9.3}. Check –2.3 + m = 7 To check your solution, substitute 9.3 for m in the original equation. –2.3 + 9.3 7 7 7 

Your Turn: Solve the equation. Check your answer. + z = 5 4 + z = Since is added to z add to both sides. The solution set is {2}. Check To check your solution, substitute 2 for z in the original equation. 

Your Turn: Solve the equation. Check your answer. The reciprocal of is . Since w is multiplied by multiply both sides by . w = 612 The solution set is {612}. Check To check your solution, substitute 612 for w in the original equation. 102 102 

Example: Application Ciro deposits of the money he earns from mowing lawns into a college education fund. This year Ciro added $285 to his college education fund. Write and solve an equation to find out how much money Ciro earned mowing lawns this year. 1 4

Example: Continued 4 is times $285  e = $285 earnings is times $285 1 4  e = $285 Write an equation to represent the relationship. The reciprocal of is . Since e is multiplied by , multiply both sides by 1 4 . 1 4  e = 285 e = $1140 The original earnings were $1140 .

Your Turn: The distance in miles from the airport that a plane should begin descending divided by 3 equals the plane’s height above the ground in thousands of feet. A plane is 10,000 feet above the ground. Write and solve an equation to find the distance from the airport at which this plane should begin descending.

Check It Out! Example 4 Continued distance divided by 3 is height d ÷ 3 = h Write an equation to represent the relationship. Substitute 10 for h. The reciprocal of is . Since d is multiplied by multiply both sides by . 1 3 3 1  d = 10 d = 30 At 10,000 feet altitude the decent should start 30,000 feet from the airport.

Joke Time How was the Roman Empire cut in half? With a pair of Caesars. What kind of lighting did Noah use for the ark? Floodlights. Who invented fractions? Henry the 1/8th.

Assignment 2.1 Exercises Pg. 91 – 93: #10 – 68 even