2. © William James Calhoun, 2001 3-1: Solving Equations with Addition and Subtraction OBJECTIVE: You need to be able to solve equations by using addition and subtraction. In math, when you say two things are equal to each other, you mean they represent the same value. We use the “=“ sign to represent this. The way to think about an equation (an expression with an = sign in it) is to think of a see-saw on a playground. The = sign is the pivot point between the two sides. = Whatever you have on this side... …must balance with what you have on this side.

3. © William James Calhoun, 2001 3-1: Solving Equations with Addition and Subtraction If I have five bricks on the left hand side of the balance... = I must add five bricks to the right hand side of the balance to make the sides balance.

4. © William James Calhoun, 2001 3-1: Solving Equations with Addition and Subtraction Now, if I take two away from the right... = What must I do to the left side to re-balance the equation? Remove two bricks. These examples have shown the effects of adding or removing values from equations. The KEY RULE is: You can add or subtract anything from either side of an equation as long as you add or subtract the SAME amount from the other side of the equation!

5. © William James Calhoun, 2001 3-1: Solving Equations with Addition and Subtraction So, we can say that 15 = 11 + 4 and then add an equal amount to both sides to get: 15 + 3 = 11 + 4 + 3. Some terms: equivalent equations - equations that have the same solution solve an equation - to isolate the variable having a coefficient of 1 on one side of the equation We will be doing a LOT of solving equations this semester and throughout your high school career. For every numbers a, b, and c, if a = b, then a + c = b + c. 3.1.1 ADDITION PROPERTY OF EQUALITY

6. © William James Calhoun, 2001 Next, ask, “What else is on the same side as the letter?” In this example, it is positive 23. EXAMPLE 1: Solve 23 + t = -16. 23 + t = -16 t = -39 +(-23) Write the equation. Ask yourself, “What is the variable I am solving for?” The answer is “t”. The opposite of positive 23 is negative 23. Therefore, we add negative 23 to both sides. (Which is the same a subtracting 23 from both sides.) This was a one-step solution. After the 23 is gone from the left hand side of the equation, we are done. Rewrite the solution equation. Remember the key to solving equations is to get the variable alone - all by itself on one side of the equation. Everything else is “moved” to the other side by using addition and subtraction. (Next section, we will use multiplication and division to isolate the variables.) 3-1: Solving Equations with Addition and Subtraction

7. © William James Calhoun, 2001 3-1: Solving Equations with Addition and Subtraction You will notice all the answers are in the format, “letter equals number.” If there is an Addition Property, then we can assume there must be a... So, we can say that 15 = 11 + 4 and then add an equal amount to both sides to get: 15 - 3 = 11 + 4 - 3. For every numbers a, b, and c, if a = b, then a - c = b - c. 3.1.2 SUBTRACTION PROPERTY OF EQUALITY Your answers should look the same - not just a number - even though the answers in the back of the book are just a number. The book does it that way to save paper and ink. You do NOT need to save the paper and ink in this manner.

8. © William James Calhoun, 2001 3-1: Solving Equations with Addition and Subtraction EXAMPLE 2: Solve 190 - x = 215. 190 - x = 215 - 190 -x = 25 x = -25 Write the equation. What is on the same side as x? Positive 190. How do you get rid of positive 190? Subtract 190. Now, we have “the opposite of x is 25.” So what is x if its opposite is 25? -25 Essentially, we just change the sign on both sides. When you solve an equation correctly, you have the value the variable was holding the place for. You can use the solution you come up with to check if your answer is correct.

9. © William James Calhoun, 2001 3-1: Solving Equations with Addition and Subtraction 190 - x = 215 x = -25The solution from Example 2 was: Go back to the original equation and plug in for x: 190 - (-25) = 215 Now see if this makes a true statement. -25 for x makes 190 - x = 215 a true statement. Therefore x = -25 is the correct solution. 215 = 215 TRUE Use the answer check on the test after you have gone through and answered all the questions.

10. © William James Calhoun, 2001 EXAMPLE 3: Solve a + 3 = -9 in two ways. METHOD 1METHOD 2 a + 3 = -9 Rewrite. Get a alone. Get rid of positive 3 by adding negative 3. Letter = number. Rewrite. Get a alone. Get rid of positive 3 by adding negative 3. Letter = number. a + 3 = -9 +(-3) a = -12 -3 3-1: Solving Equations with Addition and Subtraction Negative-negative yields positive: Now subtract from both sides: Giving: Reduce: y = -1 EXAMPLE 4: Solve

11. © William James Calhoun, 2001 EXAMPLE 5: Solve b + (-7.2) = -12.5. Rewrite. b + (-7.2) = -12.5 Write equation in easier form. Add 7.2 to both sides. Letter equals number. b - 7.2 = -12.5 b = -5.3 +7.2 3-1: Solving Equations with Addition and Subtraction Thus begins our foray into real Algebra.

12. © William James Calhoun, 2001 HOMEWORK Page 148 #15 - 33 odd 3-1: Solving Equations with Addition and Subtraction