1. Linear Equations in One Variable
6.2
Linear Equations in One Variable

2. Definitions
Terms are parts that are added or subtracted in an algebraic expression.
Coefficient is the numerical part of a term.
Like terms are terms that have the same variables with the same exponents on the variables.
Unlike terms have different variables or different exponents on the variables.

3. Properties of the Real Numbers
Associative property of multiplication
(ab)c = a(bc)
Associative property of addition
(a + b) + c = a + (b + c)
Commutative property of multiplication
ab = ba
Commutative property of addition
a + b = b + a
Distributive property
a(b + c) = ab + ac

4. Example: Combine Like Terms
8x + 4x
= (8 + 4)x
= 12x
5y  6y
= (5  6)y
= y
x + 15  5x + 9
= (1 5)x + (15 + 9)
= 4x + 24
3x y  4 + 7x
= (3 + 7)x + 6y + (2  4)
= 10x + 6y  2

5. Practice Problems: p. 306 # 22

6. Solving Equations Addition Property of Equality
If a = b, then a + c = b + c for all real numbers a, b, and c.
Find the solution to the equation
x  9 = 24.
x  =
x = 33
Check: x  9 = 24
33  9 = 24 ?
24 = 24 true

7. Solving Equations continued
Subtraction Property of Equality
If a = b, then a  c = b  c for all real numbers a, b, and c.
Find the solution to the equation
x + 12 = 31.
x + 12  12 = 31  12
x = 19
Check: x + 12 = 31
= 31 ?
31 = 31 true

8. Solving Equations continued
Multiplication Property of Equality
If a = b, then a • c = b • c for all real numbers a, b, and c, where c  0.
Find the solution to the equation

9. Solving Equations continued
Division Property of Equality
If a = b, then for all real numbers a, b, and c, c  0.
Find the solution to the equation 4x = 48.

10. General Procedure for Solving Linear Equations
If the equation contains fractions, multiply both sides of the equation by the lowest common denominator (or least common multiple). This step will eliminate all fractions from the equation.
Use the distributive property to remove parentheses when necessary.
Combine like terms on the same side of the equal sign when possible.

11. General Procedure for Solving Linear Equations continued
Use the addition or subtraction property to collect all terms with a variable on one side of the equal sign and all constants on the other side of the equal sign. It may be necessary to use the addition or subtraction property more than once. This process will eventually result in an equation of the form ax = b, where a and b are real numbers.
Solve for the variable using the division or multiplication property. The result will be an answer in the form x = c, where c is a real number.

12. Example: Solving Equations
Solve 3x  4 = 17.

13. Example: Solving Equations
Solve 21 = 6 + 3(x + 2) .

14. Example: Solving Equations
Solve 8x + 3 = 6x + 21.

15. Example: Solving Equations
Solve
False, the equation has no solution. The equation is inconsistent.

16. Example: Solving Equations
Solve
True, 0 = 0 the solution is all real numbers.

17. Practice Problems: p. 307 # 48, 50 & 58

18. Proportions
A proportion is a statement of equality between two ratios.
Cross Multiplication
If then ad = bc, b  0, d  0.

19. To Solve Application Problems Using Proportions
Represent the unknown quantity by a variable.
Set up the proportion by listing the given ratio on the left-hand side of the equal sign and the unknown and other given quantity on the right-hand side of the equal sign. When setting up the right-hand side of the proportion, the same respective quantities should occupy the same respective positions on the left and right. For example, an acceptable proportion might be

20. To Solve Application Problems Using Proportions continued
Once the proportion is properly written, drop the units and use cross multiplication to solve the equation.
Answer the question or questions asked using appropriate units.

21. Example
A 50 pound bag of fertilizer will cover an area of 15,000 ft2. How many pounds are needed to cover an area of 226,000 ft2?
754 pounds of fertilizer would be needed.

22. Homework
P. 306 # 15 – 71 eoo