3_1 Proof by Substitution

Example 1

Prove -(-5)=5 with the axioms and defintions.
Write these steps: Think these reasons:
Use law 5. a+(-a)=0 This axiom is a good start. Next use law 5 to create -(-5)).
Replace expression a with -5. (-5)+(-(-5))=0 Next indicate that this is the answer.
Underline the answer.




Example 2

Prove x+1=1+x with the axioms and defintions.
Write these steps: Think these reasons:
Use law 1. a+b=b+a This axiom is a good start, because is has the same operation and the same order of variables. Next we replace the variables.
Replace expression a with x. x+b=b+x Next replace b with 1.
Replace expression b with 1. x+1=1+x Next indicate that this is the answer.
Underline the answer.




Example 3

Prove x-1=-1+x with the axioms and defintions.
Write these steps: Think these reasons:
Use law 1. a+b=b+a This axiom is a good start, because is has the same operation and the same order of variables. Next we replace the variables.
Replace expression a with x. x+b=b+x Next replace b with -c.
Replace expression b with -c. x+(-c)=(-c)+x Next we use the definition of subtraction.
Replace expression x+(-c) with x-c. x-c=(-c)+x Next we replace c with 1.
Replace expression c with 1. x-1=(-1)+x Next indicate that this is the answer.
Underline the answer.




Example 4

Prove (3 + x) * 4 = 12 + 4 * x with the axioms and defintions.
Write these steps: Think these reasons:
Use law 9. (a+b)*c=a*c+b*c This axiom is a good start, because is has the same operation and the same order of variables. Next we replace the variables.
Replace expression a with 3. (3+b)*c=3*c+b*c Next replace b with x.
Replace expression b with x. (3+x)*c=3*c+x*c Next replace c with 4.
Replace expression c with 4. (3+x)*4=3*4+x*4 Next multiply 3 and 4.
Multiply. (3+x)*4=12+x*4 Next commute x*4.
Replace expression x*4 with 4*x. (3+x)*4=12+4*x Done.
Underline the answer.


Algebra Section 3_1
Exercises

Index