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.

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 $\frac{1}{3}$.)

Example 2

Two bags plus 10 marbles weighs the same as 5 bags and 1 marble. Find the number of marbles in each bag.

[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.

[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.

[show answer]

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

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

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.

Question 3

Find the value of n by balancing the equation algebraically.

$6n + 5 = 4n + 13$