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Solving Linear Equations
In this section we will combine all of the methods that we
have learned to solve any
linear equation containing fractions, decimals, and
parentheses. We will use all of
properties that we learned in Chapter 1 to simplify the expressions on both
sides of the
equation.
put in the form of ax + b = 0, where a and b are real numbers, a ≠ 0.
The alternate form of this equation is: A x = B or A = B x
Extend this to the general form of the equation.
General equation: Ax + B = C x + D (A, B, C, D are real numbers.).
i) Simplify the equation so that there are
only integers on
both sides
ii) Many equations have enclosures which must first be
simplified.
iii) Then solve simplified equations vertically 
using the balance beam.
Look at the coefficients of x and determine which is the larger integer (furthest to
the right on the number line ). If c > a then we will keep the variable x on that side of
the equation and keep the constant on the other side. To do this we first add
opposites on the balance beam below the equation. Look at the pattern, and then
follow the same steps through several examples
Solve simplified equations vertically  using the balance beam .
Equation: a x + b = c x + d Both sides simplified (a ,b, c, d are
integers.)
Pattern: c > a
1) Add opps :
Complete the step:
Let A = (c a) and B = (b  d)  
Then:  A > 1 is coefficient of x 
2) Multiply recip:
Example 1: Solve 3(x − 2) = 5
Procedure: 3(x − 2) = 5
1) Remove parentheses :  3x − 6 = 5  
2) Add opps:  
Complete the step:  Note: (5 + 6) = 11  
Then  3 is the coefficient of x  
3) Multiply recip:  Since  
Then  and  
4) Check in original equation:  3(x − 2) = 5 

Replace in the equation: 

or  5 = 5  x = is correct. 
Solving Linear Equations
Simplify equations that contain decimal fractions first by
multiplying by the
LCD .
Example 2:
Solve  0.05x + 0.07(100 − x) = 6.2  
Procedure:  0.05x + 0.07(100 − x) = 6.2  
Multiply LCD :  100[0.05x + 0.07(100 − x) = 6.2 ]  
5x + 7(100 − x) = 620  Distributive Property  
Remove parentheses :  5x + 700 − 7x = 620  
Collect like terms :  2x + 700 = 620  Commutative Property 
Simplify using balance beam:
1) Add opps:  Add same thing to both sides  
Complete the step:  2x =  80  (620 − 700) =  80 
2) Multiply recip:  1/2[ 2x = 80 ]  
Then  x = 40  
3) Check in original equation:  0.05x + 0.07(100 − x) = 6.2  
Replace x  0.05(40) + 0.07[100 − (40)] = 6.2  
Simplify:  2 + 7 − 2.8 = 6.2  
or   1 =  1 x = 6.2 is correct. 
Make special note of this example –
This is the type of equation that will be found in
applications involving rates, percents
and investments.
Example 3:
Solve  2(x − 3) + (x − 4) = − 5 (x +1) − 7(2x – 3)  
Procedure:  2(x − 3) + (x − 4) =  5(x +1) − 7(2x – 3)  
Remove parentheses:  2x − 6 + x − 4 =  5x − 5 − 14x+ 21  Distributive Property 
Collect like terms :  3x − 10 = 19x + 16  Commutative Property 
Simplify using balance beam:  
1) Add opps:  Note that 3 > 19 Add same thing to both sides 

Complete the step:  (3 + 19)x = +10  (3 + 19) = 22 and (16+10) = 26 
Then  22x = 26  22 is the coefficient of x 
2) Multiply recip:  
Then  
3) Check in original equation:  2(x − 3) + (x − 4) =  5(x +1) − 7(2x – 3)  
Replace x

Example 4:
Solve
Procedure: Distributive Property
1) Remove parentheses: ·
Complete the step: Associative Property
==>Solve vertically:
2) Add 0pps:
Complete the step: 8 = x or x = 8
3) Check:
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