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Elementary Algebra

Key Concepts

Elementary AlgebraKey Concepts

Key Concepts

3.1 Use a Problem-Solving Strategy

  • Problem-Solving Strategy
    1. Step 1. Read the problem. Make sure all the words and ideas are understood.
    2. Step 2. Identify what we are looking for.
    3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
    4. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation.
    5. Step 5. Solve the equation using good algebra techniques.
    6. Step 6. Check the answer in the problem and make sure it makes sense.
    7. Step 7. Answer the question with a complete sentence.
  • Consecutive Integers
     Consecutive integers are integers that immediately follow each other.
    n1stintegern+12ndinteger consecutive integern+23rdconsecutive integer . . . etc.n1stintegern+12ndinteger consecutive integern+23rdconsecutive integer . . . etc.

     Consecutive even integers are even integers that immediately follow one another.
    n1stintegern+22ndinteger consecutive integern+43rdconsecutive integer . . . etc.n1stintegern+22ndinteger consecutive integern+43rdconsecutive integer . . . etc.

     Consecutive odd integers are odd integers that immediately follow one another.
    n1stintegern+22ndinteger consecutive integern+43rdconsecutive integer . . . etc.n1stintegern+22ndinteger consecutive integern+43rdconsecutive integer . . . etc.

3.2 Solve Percent Applications

  • Percent Increase To find the percent increase:
    1. Step 1. Find the amount of increase. increase=new amount−originalamountincrease=new amount−originalamount
    2. Step 2. Find the percent increase. Increase is what percent of the original amount?
  • Percent Decrease To find the percent decrease:
    1. Step 1. Find the amount of decrease. decrease=original amount−newamountdecrease=original amount−newamount
    2. Step 2. Find the percent decrease. Decrease is what percent of the original amount?
  • Simple Interest If an amount of money, P, called the principal, is invested for a period of t years at an annual interest rate r, the amount of interest, I, earned is
    I=PrtwhereI=interestP=principalr=ratet=timeI=PrtwhereI=interestP=principalr=ratet=time
  • Discount
    • amount of discount is discount rate ·· original price
    • sale price is original price – discount
  • Mark-up
    • amount of mark-up is mark-up rate ·· original cost
    • list price is original cost + mark up

3.3 Solve Mixture Applications

  • Total Value of Coins For the same type of coin, the total value of a number of coins is found by using the model.
     number·value=totalvaluenumber·value=totalvalue where number is the number of coins and value is the value of each coin; total value is the total value of all the coins
  • Problem-Solving Strategy—Coin Word Problems
    1. Step 1.
      Read the problem. Make all the words and ideas are understood. Determine the types of coins involved.
      • Create a table to organize the information.
      • Label the columns type, number, value, total value.
      • List the types of coins.
      • Write in the value of each type of coin.
      • Write in the total value of all the coins.
    2. Step 2. Identify what we are looking for.
    3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
       Use variable expressions to represent the number of each type of coin and write them in the table.
       Multiply the number times the value to get the total value of each type of coin.
    4. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the sentence into an equation.
       Write the equation by adding the total values of all the types of coins.
    5. Step 5. Solve the equation using good algebra techniques.
    6. Step 6. Check the answer in the problem and make sure it makes sense.
    7. Step 7. Answer the question with a complete sentence.

3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

  • Problem-Solving Strategy for Geometry Applications
    1. Step 1. Read the problem and make all the words and ideas are understood. Draw the figure and label it with the given information.
    2. Step 2. Identify what we are looking for.
    3. Step 3. Name what we are looking for by choosing a variable to represent it.
    4. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
    5. Step 5. Solve the equation using good algebra techniques.
    6. Step 6. Check the answer in the problem and make sure it makes sense.
    7. Step 7. Answer the question with a complete sentence.
  • Triangle Properties For △ABC△ABC
     Angle measures:
    • m∠A+m∠B+m∠C=180m∠A+m∠B+m∠C=180
    Perimeter:
    • P=a+b+cP=a+b+c
    Area:
    • A=12bh,b=base,h=heightA=12bh,b=base,h=height
    A right triangle has one 90°90° angle.
  • The Pythagorean Theorem In any right triangle, a2+b2=c2a2+b2=c2 where c is the length of the hypotenuse and a and b are the lengths of the legs.
  • Properties of Rectangles
    • Rectangles have four sides and four right (90°) angles.
    • The lengths of opposite sides are equal.
    • The perimeter of a rectangle is the sum of twice the length and twice the width: P=2L+2W.P=2L+2W. The area of a rectangle is the length times the width: A=LW.A=LW.

3.5 Solve Uniform Motion Applications

  • Distance, Rate, and Time
    • D = rt where D = distance, r = rate, t = time
  • Problem-Solving Strategy—Distance, Rate, and Time Applications
    1. Step 1. Read the problem. Make sure all the words and ideas are understood.
       Draw a diagram to illustrate what it happening.
       Create a table to organize the information: Label the columns rate, time, distance. List the two scenarios. Write in the information you know.
    2. Step 2. Identify what we are looking for.
    3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
       Complete the chart.
       Use variable expressions to represent that quantity in each row.
       Multiply the rate times the time to get the distance.
    4. Step 4. Translate into an equation.
       Restate the problem in one sentence with all the important information.
       Then, translate the sentence into an equation.
    5. Step 5. Solve the equation using good algebra techniques.
    6. Step 6. Check the answer in the problem and make sure it makes sense.
    7. Step 7. Answer the question with a complete sentence.

3.6 Solve Applications with Linear Inequalities

  • Solving inequalities
    1. Step 1. Read the problem.
    2. Step 2. Identify what we are looking for.
    3. Step 3. Name what we are looking for. Choose a variable to represent that quantity.
    4. Step 4. Translate. Write a sentence that gives the information to find it. Translate into an inequality.
    5. Step 5. Solve the inequality.
    6. Step 6. Check the answer in the problem and make sure it makes sense.
    7. Step 7. Answer the question with a complete sentence.
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