Multi-Step Equations
Lesson 9
We will be working with multi-step equations which will eventually turn into two step equations so it is very important for your student to be able to solve for two step equations.
We will introduce our SOLVE problem: Leslie and two friends went out to lunch. They each ordered a 2 dollar drink and a hamburger meal. They did not order the 3 dollars dessert because they were so full. Their bill came to 27 dollars. How much did each hamburger meal cost? We are going to study the problem; in “S” we underline the questions. Our question is, how much did the hamburger meal cost? We also have to ask the question, what is this problem asking me to find. This problem is asking me to find, the cost of each hamburger meal.
When we have like terms, such as 3T plus 4T, they can be combined. We can represent 3T with 3 T’s and we can represent 4T with 4 T’s. 3 plus 4 T’s is equal to 7T.
When we have like terms such as 4P minus P, we can represent this pictorially. We can represent our 4P with 4 P’s. Minus P means we want to take away a P. When we take a P we are left with 3 P’s. So, 4P minus P equals 3P.
When 4d plus 2c plus 2d we can make a pictorial representation. We can represent our 4d with 4 d’s, our 2C with 2 C’s, and our 2d with 2 d’s. We cannot combine D’s and C’, but we can move the order so that we put these 2 d’s with the 4 d’s. Creating 6D plus 2C. 4D plus 2C plus 2D is equal to 6D plus 2C.
We want to model the problem 4X plus 2X equals 12. We will represent our 4X with 4 cups, we will represent our 2X with 2 cups, and we will represent our 12 with 12 yellow integer chips, because 12 is positive and yellow is positive. 4X plus 2X or 4 cups plus 2 cups is just 6 cups. We have 6 cups equal to 12, so we are going to determine what the value would be if we split our 12 into the 6 cups. The value of one cup is 2 yellows or 2. Since C equals 2 then in our problem X will equal 2. To check our answer we go back to our original equation. Of 4C plus 2C equals 12. We will replace each cup with 2 yellows, seeing that 12 equals 12, our answer checks. If we want to solve our equation pictorially, we can represent 4X with 4 X’s, and we can represent our 2X with 2 X’s. Which is equal to 12 and we represent 12 with 12 Y’s. We see that we have 6 X’s and if we wanted to divide our Y’s evenly to each X. One X would be equal to 2 Y’s. So the value for X is 2. For the problem 4X plus 2X equals 12, we know that 4X plus 2X equals 6X and when we have 6X and we want to isolate the variable, we have multiply by 6, so the opposite of multiply by 6 is divide by 6. We divide both side by 6 and X equals 2. We can check our answer by rewriting the problem, 4X plus 2X equals 12. If X equals 2, we substitute in a 2 for X, 4 times 2 is 8 and 2 times 2 is 4, 8 plus 4 is 12, so our answer checks.
To model a problem 2X plus 2 equals X plus 5; we will have 2 cups and 2 yellows on the left hand side, and we will have 1 cup and 5 yellows on the right hand side. This problem is different than others we have worked but because it has a variable on both sides. We know we cannot isolate a variable if it is on both sides, so we must move one of these cups. If we wanted to take away 2 cups from the left hand side, it is impossible to take away 2 cups from the right hand side. We could however, take away one cup from the right hand side and in order to stay balanced, one cup from the left hand side. We subtracted one cup, or subtracted X, from each side. Leaving us with a cup plus 2 equals 5. From here we want to isolate our one cup, so we take away 2 yellows from each side. And our cup, or X the variable, is equal to 3. We can check our answer by recreating the original problem, of 2X plus 2 equals X plus 5. We said that the answer for 1 cup was 3. So we have to replace each cup with 3 chips. When we replace each cup with 3 chips we see we have 8 on the left hand side and 8 on the right hand side. So our answer is correct. To model this problem pictorially, we would have 2 X’s and 2 Y’s on the left hand side and 1 X and 5 Y’s on the right hand side. We took away a X or subtracted a X, so we draw a line through it, leaving us with X plus 2. We have to remove those 2 Y’s, so now we can see that our answer is X equals 3 Y’s. Or X equals 3. If we look at the abstract problem, we took away X from both sides. The process of taking away us subtraction, we subtract X from both sides, 2 X minus X is X plus 2 and X minus X cancels leaving us with just 5. Here in order to isolate our X, here we have X plus 2 and the opposite of plus 2 is subtract 2. You subtract 2 from both sides and X equals 3. We can check our answer by rewriting the original equation; we plug 3 in for X. 2 times 3 is 6; 6 plus 2 is 8; and 3 plus 5 is 8. So our answer checks.
To complete our SOLVE problem we have already decided that the question is how much did each hamburger meal cost. This problem is asking me to find the cost of each hamburger meal. In “O” we identify the facts. Leslie and two friends went out to lunch, they each ordered a $2 drink, and a hamburger meal, they did not order the $3 dessert because they were so full, their bill came to 27 dollars. We must decide which facts are necessary and unnecessary. Leslie and two friends went out to lunch, this fact is necessary; Leslie and 2 friends means that we have 3 people. They each ordered a $2 drink, this fact is also necessary; and a hamburger meal; they did not order the $3 dessert because they were so full, this fact is unnecessary. Their bill came to 27 dollars In “L” we are going to line up our plan. We choose our operations, which are multiplication. Our plan is to create a equation relating the number of people multiplied by the cost of the hamburger meal, M will represent the hamburger meal, plus the cost of a drink multiplied by the number of people equal to the total bill. In “V” we will estimate our answer in verifying our plan with action. We know that their bill came to 27 dollars, and we are going to have to take some money off for the drink and then we are going to divide that 3 ways. If you divided the 27 dollars 3 ways first, 27 is close to 30, so we know that that’s 10 dollars, 30 divided by 3 is 10 dollars and then they have the cost of he drink, so you can take 2 dollars off of that. A good estimate could be less than 8 dollars. Some students will be able to say that their hamburger meal must be less than 27 dollars and some students may say that their hamburger meal would be more than 5 dollars. There are lots of different estimates. When we carry out our plan we write our equation. 3M represents the cost of the hamburger meal times the number of people that had a hamburger meal. Plus 3 times 2 which is our 3 friends times the 2 dollar drinks equal to 27 dollars. 3M plus 6 equals 27, because 3 times 2 is 6; the first step to isolating the variable is to subtract 6 from both sides, the opposite of plus 6 is to subtract 6; this leaves us with 3M equal to 21. Here 3 times M in order to isolate the M, the opposite of multiply by 3 is divide by 3; M equals 21 divided by 3 which is 7. In “E” we examine our results. Does your answer make sense? We go back to the question, how much did each hamburger meal cost? Could a hamburger meal cost 7 dollars? Yes, so our answer makes sense. Is your answer reasonable? We look at our estimate, one of the estimates we had was more than 5 dollars. Is your answer accurate? We can check for accuracy by rewriting the problem and checking our work. Our value for M is 7. 21 plus 6 is equal to 27; so our answer is correct. And now we want to write our answer as a complete sentence. The hamburger meal cost 7 dollars each.
To close our lesson we will review the essential questions. Number 1, what are like terms? Like terms are terms that have the same variable or the same last name; and example is 3X and 2X. Number 2, how do you solve and equation with variables on both sides of the equals sign? You first use and inverse operation to combine the like terms, then solve the two or one step equations which is left.