Solving Linear Equations
Techniques for Solving Linear Equations
Writing down every step when solving an equation is not always necessary.
Solving an equation is often part of a larger problem, and anything that we can
do to make the process more efficient will make solving the entire problem
fastar and easier. For example, we can combine some steps.
Combining Steps 

Writing Every Step 
4x  5 
= 23 

4x  5 
= 23 
4x 
= 28 
Add 5 to each side. 
4x  5 + 5 
= 23 + 5 
x 
= 7 
Divide each side by 4. 
4x 
= 28 








x 
= 7 
The same steps are used in each of the solutions. However, when 5 is added
to each side in the solution on the left, only the result is written. When
each side is divided by 4, only the result is written.
The equation x = 5 says that the additive inverse of x is 5. Since the
additive inverse of 5 is 5, we conclude that x is 5. So instead of
multiplying each sideof x = 5 by 1, we solve the equation as follows:
x 
= 5 

x 
= 5 
Additive inverse property 
Sometimes it is simpler to isolate x on the righthand side of the equation:
3x + 1 
= 4x  5 

6 
= x 
Subtract 3x from each side and add 5 to each side. 
You can rewrite 6 = x as x = 6 or leave it as is. Either way, 6 is the
solution.
For some equations with fractions it is more efficient to multiply by a
multiplicative inverse instead of multiplying by the LCD:





Multiply each side by
the reciprocal of

x 


