3 1 Solving Equations with Addition and Subtraction

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. © 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. © 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! © William James Calhoun, 2001

3 -1: Solving Equations with Addition and Subtraction 3. 1. 1 ADDITION PROPERTY OF EQUALITY For every numbers a, b, and c, if a = b, then a + c = b + c. 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. © William James Calhoun, 2001

3 -1: Solving Equations with Addition and Subtraction EXAMPLE 1: Solve 23 + t = -16. Write the equation. Ask yourself, “What is the variable I am solving for? ” The answer is “t”. Next, ask, “What else is on the same side as the letter? ” In this example, it is positive 23. 23 + t = -16 +(-23) t = -39 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. ) © 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. ” 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. If there is an Addition Property, then we can assume there must be a. . . 3. 1. 2 SUBTRACTION PROPERTY OF EQUALITY For every numbers a, b, and c, if a = b, then a - c = b - c. So, we can say that 15 = 11 + 4 and then add an equal amount to both sides to get: 15 - 3 = 11 + 4 - 3. © William James Calhoun, 2001

3 -1: Solving Equations with Addition and Subtraction EXAMPLE 2: Solve 190 - x = 215. 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. 190 - x = 215 - 190 -x = 25 x = -25 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. © William James Calhoun, 2001

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

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

3 -1: Solving Equations with Addition and Subtraction EXAMPLE 5: Solve b + (-7. 2) = -12. 5. Rewrite. Write equation in easier form. Add 7. 2 to both sides. Letter equals number. b + (-7. 2) = -12. 5 b - 7. 2 = -12. 5 +7. 2 b = -5. 3 Thus begins our foray into real Algebra. © William James Calhoun, 2001