## Converting Bags and Marbles into Equations

 In this lesson we will begin by solving a puzzle using bags of marbles. Then we will write the algebraic equation represented by the bags and marbles and show that the solution satisfies the equation.

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First, let's solve the following puzzle to determine the number of marbles in each bag.

 One bag and nine marbles on the left weigh the same as four bags and three marbles on the right. How many marbles are in each bag?

To solve this problem, first remove one marble bag from each side, then remove three marbles from each side, which leaves three marble bags on the right and six marbles on the left. Now it is easy to see that each marble bag contains two marbles.

The bags and marbles are a model to make it easier for you to think about algebraic equations. Since the two sides of the balance weigh the same amount, we can write an equation to represent this situation.

 The model... The algebraic equation... $b + 9 = 4b + 3$ Verify that b = 2 is the correct solution... Evaluate the equation for $b = 2$. $\matrix{b+9&=&4b+3\\2+9&=&(4)(2)+3\\11&=&8+3\\11&=&11\\&\large\surd~&}$ Since b = 2 causes both sides of the equation to equal each other, 2 is the correct solution.

 Example 1 Find the number of marbles in each bag. Write the equation represented by the model. Verify your answer using substitution.

 Find the answer... Write the equation and check the answer... [show answer] Evaluate the equation for $b = 5$. $\matrix{b+12&=&3b+2\\5+12&=&(3)(5)+2\\17&=&17\\&\large\surd~&}$

 Example 2 Find the number of marbles in each bag. Write the equation represented by the model. Verify your answer using substitution.

 Find the answer... Write the equation and check the answer... [show answer] Evaluate the equation for $b = 5$. $\matrix{3b+2&=&2b+7\$$3)(5)+2&=&(2)(5)+7\\15+2&=&10+7\\17&=&17\\ &\large\surd~&}$ # Self-Check  Question 1 Maria claims that the solution to this puzzle is each bag contains 3 marbles. Is she correct? [show answer] $b=3$ $2b+5=3b+2$ $(2)(3)+5=(3)(3)+2$ $6+5=9+2$ $11=11$ Maria is correct!  Question 2 Martin says that this puzzle can be represented algebraically as $$$2x+9=4x+1\($ Is he correct? [show answer]   Yes he is correct. (By the way, the solution is $b = 4$)

 Question 3 Find the number of marbles in each bag. Write the algebraic equation that this puzzle models. [show answer]   $b=3$ $3b+8=5b+2$