1-2 Solving Linear Equations

Question 1

What is meant by a “solution”?

Answer 1

a value for the variable that makes the equation a true statement

Question 2

What are characteristics of variables?

Answer 2

an unknown quantity

a quantity that can vary

usually represented by a letter of the alphabet

Question 3

How can one fraction or more than one fraction with the same denominator be eliminated when solving equations?

Answer 3

multiplying EVERY term by denominator
__________________________
[see 2. clearing fractions]

Question 4

How are consecutive odd integers solved?

Answer 4

1) define the variable
Let x = first odd consecutive integer
x + 2 = second odd consecutive integer
x + 4 = third odd consecutive integer

2) write equation
x + (x + 2) + (x + 4) = sum of three consecutive odd integers

3) solve

4) calculate the three consecutive odd integers by substituting the value for x obtained from solving the equation

5) identify the three consecutive odd integers
_____________________________
[see 3. consecutive integers]

Question 5

How are consecutive even integer problems solved?

Answer 5

1) define the variable
Let x = first consecutive even integer
x + 2 = second consecutive even integer
x + 4 = third consecutive even integer

2) write the equation
x + (x + 2) + (x + 4) = sum of first three consecutive even integers

3) solve

4) calculate the three consecutive even integers by substituting the value for x obtained from solving the equation

5) identify the three consecutive even integers
_____________________________
[see 3. consecutive integers]

Question 6

How are consecutive integers solved?

Answer 6

1) define the variable
Let x = first consecutive integer
x + 1 = second consecutive integer
x + 2 = third consecutive integer

2) write equation
x + (x + 1) + (x + 2) = sum of three consecutive integers

3) solve

4) calculate the three consecutive integers by substituting the value for x obtained from solving the equation

5) identify the three consecutive integers
_____________________________
[see 3. consecutive integers]

Question 7

How to check the solution of an equation?

Answer 7

Substitute the value of the variable obtained from solving the equation.

IF the number that was substituted for the variable in the equation makes a true statement, the number is a solution to the equation.

IF the number that was substituted for the variable in the equation makes a false statement, the number is not a solution to the equation.

Question 8

How are mixture problems solved?

Answer 8

1) define the variable
Let x = amount of first solution
Let total amount of mixture - x = amount of second solution

2) percents must be converted to decimals by moving the decimal two places to the left

3) write the equation
(% of first solution as decimal)(x) +
(% of second solution as a decimal)(total amount of mixture - x)
= (total amount of mixture)(% of mixture as a decimal)

4) solve

5) identify how much of first solution, which is the value for x

6) identify how much of second solution will be needed by taking the total amount and subtracting the value of x from it
_______________
[see 4. mixture]

Question 9

How is the Distributive Property simplified?

Answer 9

1) multiply outside term and first inside term

2) then multiply outside term and second inside term

EX: a(b + c) = ab + ac
_______________________________________
[see 1. using the Distributive Property]

Question 10

How are fractions with different denominators be eliminated when solving equations?

Answer 10

1) calculate the least common denominator (LCD)

2) multiply EVERY term by the least common denominator (LCD)
_________________________
[see 2. clearing fractions]