Moving from bags to marbles to using ALGEBRA

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...

[show answer]

$\matrix{2m+10&=&5m+1\\-2m&&-2m\\&&\\10&=&3m+1\\-1&&-1\\&&\\9&=&3m\\\overline{3}&&\overline{3}\\&&\\3&=&m}$

$\matrix{2m+10&=&5m+1\\(2)(3)+10&=&(5)(3)+1\\6+10&=&15+1\\16&=&16\\&\large\surd~&}$

 


 

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...

[show answer]

 $\matrix{b+8&=&5b\\-b&&-b\\&&\\8&=&4b\\\overline{4}&&\overline{4}\\&&\\2&=&b}$

 $\matrix{b+8&=&5b\\2+8&=&(5)(2)\\10&=&10\\&\large\surd~&}$

 


 

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...

[show answer]

 $\matrix{3b+2&=&b+10\\-b&&-b\\&&\\2b+2&=&10\\-2&&-2\\&&\\2b&=&8\\\overline{2}&&\overline{2}\\&&\\b&=&4}$

 $\matrix{3b+2&=&b+10\\(3)(4)+2&=&4+10\\12+2&=&14\\14&=&14\\&\large\surd~&}$

 

 


 

Self-Check


Question 1

Write the equation for the model, and then solve it algebraically.

 

[show answer]


Question 2

Write the equation for the model and then solve it algebraically.

[show answer]


Question 3

Find the value of n by balancing the equation algebraically.

$6n + 5 = 4n + 13$

[show answer]


 

 

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