In this lesson you will continue using the bags and marbles model to help you in solving equations. This time, however, you will record each step with an algebraic equation and a verbal explanation. 
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Example 1 One bag plus 9 marbles weighs the same as 4 bags and 3 marbles. Find the number of marbles in each bag. 
Model 
Algebraic equation 
Verbal 
$\matrix{m + 9 &=& 4m + 3}$ 
Expressing the puzzle. 

$\matrix{m + 9 &=& 4m + 3\\3&&3\\&&\\m + 6 &=& 4m}$ 
Additive property of equality. (We added –3 to both sides.) 

$\matrix{m + 6 &=& 4m\\m&&m\\&&\\6 &=& 3m}$ 
Additive property of equality. (We added –1m to both sides.) 

$\matrix{6 &=& 3m\\\overline{3}&&\overline{3}\\&&\\2 &=& m}$ 
Multiplicative property of equality. (We multiplied both sides by $\frac{1}{3}$.) 
According to the model and the algebraic equations, $m=2$. However, it is important to check it by evaluating the original equation by subsituting 2 wherever we see an m.
$\matrix{m+9&=&4m+3\\2+9&=&(4)(2)+3\\11&=&8+3\\11&=&11\\&\large\surd~&}$
Example 2 Two bags plus 10 marbles weighs the same as 5 bags and 1 marble. Find the number of marbles in each bag. 
Model  Algebraic equations  Write the equation and check the solution... 
$\matrix{2m+10&=&5m+1\\2m&&2m\\&&\\10&=&3m+1\\1&&1\\&&\\9&=&3m\\\overline{3}&&\overline{3}\\&&\\3&=&m}$ 
$\matrix{2m+10&=&5m+1\\(2)(3)+10&=&(5)(3)+1\\6+10&=&15+1\\16&=&16\\&\large\surd~&}$ 
Example 3 One bag plus 8 marbles weighs the same as 5 bags. Find the number of marbles in each bag. 
Model  Algebraic equations  Write the equation and check the solution... 
$\matrix{b+8&=&5b\\b&&b\\&&\\8&=&4b\\\overline{4}&&\overline{4}\\&&\\2&=&b}$ 
$\matrix{b+8&=&5b\\2+8&=&(5)(2)\\10&=&10\\&\large\surd~&}$ 
Example 4 Three bags plus 2 marbles weighs the same as 1 bag plus 10 marbles. Find the number of marbles in each bag. 
Model  Algebraic equations  Write the equation and check the solution... 
$\matrix{3b+2&=&b+10\\b&&b\\&&\\2b+2&=&10\\2&&2\\&&\\2b&=&8\\\overline{2}&&\overline{2}\\&&\\b&=&4}$ 
$\matrix{3b+2&=&b+10\\(3)(4)+2&=&4+10\\12+2&=&14\\14&=&14\\&\large\surd~&}$ 
Question 1 Write the equation for the model, and then solve it algebraically.

Question 2 Write the equation for the model and then solve it algebraically. 
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Question 3 Find the value of n by balancing the equation algebraically. $6n + 5 = 4n + 13$ 
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