Algebra 1: Simplifying Expressions, Solving Equations, and Applying Formulas - Prof. Patri, Study notes of Algebra

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Section 2.1: Simplifying Expressions Definition: Two or more terms with the same variable part are called similar or like terms. Coefficient – number in front of the variable To combine ‘like terms’ we add or subtract the coefficient and leave the variable and exponent alone. Examples:

1. 12 y^2 ^3 y^7 ^4 y^2 ^2 y^10
2. ^2 x^ ^2 (^7 x^4 )

3. 4 ^ x^2 ^ ^3 x^5 

The Value of an Expression is the solution after you substitute a number in for the variable and simplify. Examples:

1. Find the value of (^) n^2  9 n 2 when n=  3.
2. Find the value of 6a + 5b – 7 when a = 2 and b = -3.

Section 2.2: Addition Property of Equality Definition: The solution set is the set of all numbers that make an equation a true statement. Is 3 a solution to 5x – 6 =9? Is –4 a solution to 6y – 4=-20? Addition Property of Equality : You can add or subtract the same number on both sides of an equation without changing the value of the equation. A = B A+C = B+C for any real number C Examples

1. x – 7 = 18
2. 8 1 d 3 2  
3. 10 – 3x + 4x = 20

Section 2.3: Multiplication Property of Equality Multiplication Property of Equality : You can multiply or divide the same number on both sides of an equation without changing the value of the equation. A = B A(C) = B(C) for any real number C Examples:

1. -8x=
2. z^12 3 2 
3. 4 6 a 
4. -4a – 3 = -
5. 8w + 7 – 3w =3w – 8

6. 3(x + 4) = 8x – 13
7. 5 2 x 5 1 5 2 x 3 1    You can eliminate fractions by multiplying the entire equation by the LCD. Re-do the previous example by eliminating the fractions first. 5 2 x 5 1 5 2 x 3 1   

Section 2.4: Solving Linear Equations Strategy for Solving Equations with One Variable

1. Get rid of parentheses by using the distributive property.
2. Get rid of fractions by multiplying by the LCD (or decimals by multiplying by a power of 10).
3. Simplify each side of the equation by combining like terms.
4. Get variables on one side of the equation and constant terms on the other side using the addition property of equality.
5. Solve for the specified variable by using the multiplication property of equality.
6. Check your solution. Examples:
7. –25 = 5(3x+4)
8. (x^6 ) 2 1 (x 4 ) 3 1   
9. 0.4x – 0.1 =0.7 – 0.3(6 – 2x)

4 21 x 6 1 4 3 x 3 2   

Section 2.5: Formulas Definition: In mathematics, a formula is an equation that contains more than one variable. (It usually describes some known relationship.) Examples: Example: The area of a triangle is 15 square feet. You know the base is 6 feet. Find the height of this triangle. Example: Find x when y=5 in the formula 5x–4y=20. Example: Solve 5x – 4y = 20 for y. Example: Solve for y: 1 3 y 7 x  

More Examples:

1. What number is 75% of 40?
2. What percent of 28 is 21?
3. 360 is 12% of what number? Section 2.6: Applications Problem Solving Strategy
4. Read and Understand the Problem
5. Assign a Variable to your Unknown
6. Translate in an Equation
7. Solve the Equation
8. Write your answer in a complete sentence
9. Check your Solution Number Problems The difference of 8 times a number and 4 is the same as 3 times the same number minus 14. Find the number.

Age Problems: OMIT Geometry One side of a triangle is 2 times the 1st^ side. The third side is 7 more than the 1st^ side. The perimeter of the triangle is 43 cm. Find all 3 sides of the triangle. Coin Problems My Dear Aunt Sally has $2.65 cents in nickels and dimes that has been left in the bottom of her purse. She knows that she had 5 more nickels than dimes. How many of each coin does Aunt Sally have in the bottom of her purse?

Section 2.7: More Applications Interest Problems Page 157, # Geometry In a right triangle, one acute angle is half the size of the other acute angle. Find all three angles. Miscellaneous Page 159, #

Section 2.8: Linear Inequalities Solving Inequalities is very similar to solving equations. Addition Property of Inequality If a, b, and c are real numbers, then a < b a + c < b + c The same is true for >. (You can add the same # to both sides.) Examples:

1. x+3<
2. x^ ^6 ^3 Multiplication Property of Inequality If a, b, and c are real numbers and a < b, then when c is POSITIVE, ac<bc NEGATIVE, ac>bc. Examples:
3. 4 y^ ^20

2.  7 a  28
3. 3x+12>5x – 4
4. 8 2 w  
5. 2 (^3 x^1 )^5 ^8 x^7