Involving One Operation Lesson 2-1 Solving Equations Involving One Operation

Standards California Preparation for 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable, and provide justification for each step. Also covered: 2.0

The two sides on a balanced scale must be equal to each other A Question of Balance The two sides on a balanced scale must be equal to each other E + 6 = 11 E = 5 What does the Egg weigh?

What happens if we change one of the sides of a balanced equation? A Question of Balance What happens if we change one of the sides of a balanced equation? 8 + 3 + 1 8 + 3 + 1 11 Then it is not balanced!

We need to make the same change to the other side! A Question of Balance What happens if we change one of the sides of a balanced equation? 11 + 1 8 + 3 + 1 Then it is not balanced! We need to make the same change to the other side!

A Question of Balance 8 + 3 + 1 11 + 1 What happens if we change one of the sides of a balanced equation? 8 + 3 + 1 11 + 1 We need to make the same change to the other side! We need to make the same change to the other side!

To solve an equation means to find every number that makes the equation true. We do this by adding or subtracting to each side of the equation … but always keep it balanced!

Multiply by x. Divide by x. 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. Inverse Operations Add x. Subtract x. Multiply by x. Divide by x.

Let’s go back to the balance In the equation, 7 added to a number gives 15… Solving the equation means, finding the value of the variable that makes the equation true. Let’s go back to the balance

x +7 - 7 15 - 7 Subtract 7 from both sides Simplify both sides

Subtraction Property of Equality x 8 Subtract 7 from both sides Now we know the value of x Simplify both sides

Subtraction Property of Equality Subtracting the same number from each side of an equation produces an equivalent equation x 8 So the solution goes like this… x + 7 = 15 x + 7 – 7 = 15 – 7 Subtract 7 from both sides Simplify both sides x = 8 Now we know the value of x

Solving equations using Addition and Subtraction

The Box Method Let’s try to solve backwards… x – 11 = 13 - 11 x 13 + 11 24 13

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

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

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

Now you try x + 4 = 10 2) 12 = x + 8 y + 7.1 = 1.2 4) 5.3 = y – 2 5) n – 6 = -9 6) -3 = w + 1 -4 -4 -8 -8 x = 6 4 = x +2 - 7.1 - 7.1 +2 y = -5.9 7.3 = y +6 +6 -1 -1 -4 = w n = -3

Solving equations using Multiplication and Division

Solve each equation. Check your answer. Since y is multiplied by 0.5, divide both sides by 0.5 to undo the multiplication. 5y = –85 5y = –85 The solution set is {–20}. 5 5 x5 y -85 y = -17 ÷5 -17 -85

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

In some equations, the solution is obvious. x – 7 = 12 5n = 35 x = 19 n = 7 = 3 20 + h = 41 h = 21 c = 24 We can simply work the operation backwards in our head to get the answer.

But in other equations, the solution is not so obvious. We have to know what operation(s) must be done to solve it, and work it out carefully.

But in other equations, the solution is not so obvious. You have to do the inverse operation to both sides to get the variable by itself But in other equations, the solution is not so obvious. The inverse operation of subtraction is addition The inverse operation of addition is subtraction Addition property of equality Subtraction property of equality The inverse operation of division is multiplication The inverse operation of multiplication is division Multiplication property of equality Division property of equality

Solve the equations below… 4x = 16 2) x 3 4) y 2 5) 6n = 42 6) -21 = 3w 11.2 = x = 4 x = 33.6 x 3 12 = 16 = x = 36 x = 7 w = - 7

Solving Equations by Using Opposites or Reciprocals Solve each equation. The reciprocal of is . Since w is multiplied by multiply both sides by . The solution set is {–24}.

Lesson Quiz Solve each equation. 1. r – 4 = –8 3. 2. 5. 6. This year a high school had 578 sophomores enrolled. This is 89 less than the number enrolled last year. Write and solve an equation to find the number of sophomores enrolled last year. –4 2. 2.8 4. 8y = 4 40 s – 89 = 578; s = 667