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. 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. 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! 1
ADDITION PROPERTY OF EQUALITY For every numbers a, b, and c, if a b, then a + c b + c. 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 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. 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. 23 + t -16-23 -23 t -39 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. 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... SUBTRACTION PROPERTY OF EQUALITY For every numbers a, b, and c, if a b, then a - c b - c. 2
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 - 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. 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) 215 215 215 TRUE -25 for x makes 190 - x 215 a true statement. Therefore x -25 is the correct solution. EXAMPLE 3: Solve a + 3-9. Rewrite. a + 3-9 Get a alone. Get rid of positive 3 by adding negative 3. Letter number. -3-3 a -12 3 4 EXAMPLE 4: Solve y 7 7 3 4 Negative-negative yields positive: y + 7 7 3 3 Now subtract from both sides: 7 7 7 Giving: y 7 Reduce: y -1 3
EXAMPLE 5: Solve b + (-7.2) -12.5. Rewrite. b + (-7.2) -12.5 Write equation in easier form. b - 7.2-12.5 Add 7.2 to both sides. +7.2 +7.2 Letter equals number. b -5.3 To solve equations using multiplication and division. You can add and subtract to and from both sides of an equation to maintain balance. The same is true for multiplication and division. If you double or triple one side of the equation (multiply by 2 or 3) then you must also double or triple the other side to maintain balance. If you cut one side in half (divide by 2) then you must do the same to the other side. After asking how the letter and number are hooked up, we will be entering into the multiplication & division mode of solving. Remember: WHATEVER YOU DO TO ONE SIDE OF AN EQUATION YOU MUST ALSO DO TO THE OTHER SIDE! Here are the official properties: MULTIPLICATION PROPERTY OF EQUALITY For any numbers a, b, and c, if a b, then ac bc. DIVISION PROPERTY OF EQUALITY a b For any numbers a, b, and c, if a b, then. c c 4
EX1β EXAMPLE 1α: Solve g 5. 24 12 EXAMPLE 1β: Solve each equation. 3 a 2 b a. b. c. 4 12 7 14 3 c 5 30 EX3β EXAMPLE 2α: Solve 12x 180. Write the equation. What is the letter? x What is on the same side? 12 Hooked up by? multiplication Undo with? divide Divide both sides by 12. 12x 180 12 12 x 15 5
EXAMPLE 2β: Solve each equation. a. -5t 60 b. 15 6n c. -3v -129 EXAMPLE 3α: Solve. 1 1 3 p 2 4 2 EXAMPLE 3β: Solve each equation. a. 1 1 3 t 5 b. 3 1 1 2 1 m 4 c. 5 y 4 2 4 4 2 5 20 6