upcat math

    Cards (71)

    • Equation
      A statement that two mathematical expressions are equivalent or equal
    • Solutions or roots
      The values of the unknown that make the equation true
    • Solving the equation
      Finding the solution
    • Kinds of equations
      • Identity equation
      • Conditional equation
      • Equivalent equations
      • Inconsistent equation
      • Consistent equation
    • Identity equation
      An equation that is true for any number substituted to the variable
    • Conditional equation
      An equation that is true only for certain values of the unknown
    • Equivalent equations

      Two equations with exactly the same solutions
    • Inconsistent equation
      An equation that has no solution
    • Consistent equation
      An equation that has a solution
    • Determine whether the given equation is an identity or a conditional equation
    • Linear equation in one variable

      An equation that can be written in the form ax + b = 0 where a and b are real numbers and a ≠ 0
    • Examples of linear equations

      • 2x1 = 0
      • -5x = 10 + x
      • 3x + 8 = 2
    • Types of nonlinear equations
      • Contains the square of the variable
      • Contains the reciprocal of the variable
      • Contains the square root of the variable
    • Steps in solving a linear equation in one variable
      1. Simplify algebraic expressions
      2. Gather variable terms
      3. Isolate the variable
      4. Check the solution
    • Example of solving a linear equation
      • 2(x-1)+3=x-3(x+1)
    • Steps in solving rational equations
      1. Determine excluded values
      2. Multiply by the LCD
      3. Solve the resulting linear equation
      4. Eliminate extraneous solutions
    • Rational equation
      An equation that contains one or more rational expressions
    • Extraneous solution

      Solutions that satisfy a transformed equation but do not satisfy the original equation
    • Linear equations are solved by simplifying, gathering terms, isolating the variable, and checking the solution
    • Rational equations are solved by determining excluded values, multiplying by the LCD, solving the resulting linear equation, and eliminating extraneous solutions
    • Today's objectives
      • Develop mathematical models
      • Solve application problems
      • Solve number problems
      • Solve digit problems
      • Solve geometric problems
      • Solve money and coin problems
    • Steps in solving word problems
      1. Read and analyze
      2. Make a diagram
      3. Determine the unknown
      4. Set up an equation
      5. Solve the equation
      6. Check the solution
    • Number problems
      • Find three consecutive odd integers
      • Find two consecutive even integers
    • Geometry problems
      • A rectangle with the same area as a square
      • Two circles with a specific ratio of circumferences
    • Digit problems
      • Integer between 10 and 100
      • Two-digit number equal to 9 times the sum of its digits
      • Sum of digits of a two-digit number is 11
    • Money and coin problems

      • A change purse with equal numbers of coins totaling $1.44
    • The ratio of the circumferences is 2:1
    • There are problems related to digits, money, investment, age, mixture, uniform motion, work, and clock
    • Problem-solving procedure
      1. Read and analyze the problem
      2. Make a diagram or sketch
      3. Determine the unknown quantity
      4. Set up an equation
      5. Solve the equation
      6. Check the solution
    • Types of problems
      • Digit problems
      • Money and coin problems
      • Investment problems
      • Age problems
      • Mixture problems
      • Uniform motion problems
      • Work problems
      • Clock problems
    • Digit problems
      • In an integer between 10 and 100, the unit’s digit is 3 greater than the ten’s digit. Find the integer, if it is 4 times as large as the sum of its digits.
      • A certain two digit number is equal to 9 times the sum of its digits. If 63 were subtracted from the number the digits would be reversed. Find the number.
      • The sum of the digits of a two-digit number is 11. If we interchange the digits then the new number formed is 45 less than the original. Find the original number.
    • Money and coin problems

      • A change purse contains an equal number of pennies, nickels and dimes. The total value of the coins is $1.44. How many of each type does the purse contain?
      • Mary has $3.00 in nickels, dimes and quarters. If she has twice as many dimes as quarters and five more nickels than dimes, how many coins of each type does she have?
    • Investment problems
      • An ambitious 14-year old has saved $1,800 from chores and odd jobs around the neighborhood. If he puts this money into a CD that pays a simple interest rate of 4% a year, how much money will he have in his CD at the end of 18 months?
      • Theresa earns a full athletic scholarship for college, and her parents have given her the $20,000 they had saved to pay for her college tuition. She decides to invest that money with an overall goal of earning 11% interest.
    • Age problems
      • A father is four times as old as his daughter. In 6 years, he will be three times as old as she is now. How old is the daughter now?
      • A movie star posed a riddle about his age in relation to his daughter’s age.
    • Mixture problems

      • A mechanic is working on the coolant system of a vehicle with a capacity of 11.0 liters. Currently the system is filled with coolant that is 45% ethylene glycol. How much fluid must be drained and replaced with 100% ethylene glycol so that the system will be filled with coolant that is 60% ethylene glycol?
      • For a certain experiment, a student requires 100 ml of a solution that is 8% HCl. The storeroom has only solutions that are 5% and 15% HCl.
    • Uniform motion problems
      • You and your roommate decided to take a road trip to the beach one weekend. You drove all the way to the beach at an average speed of 60 mph.
      • A Cessna 150 averages 150 mph in still air.
    • Work problems
      • Connie can clean her house in 2 hours. If Alvaro helps her, they can clean the house in 1 hour and 15 minutes together.
      • Next-door neighbors Bob and Jim use hoses from both houses to fill Bob’s swimming pool.
    • Clock problems

      • What time after 8 o’ clock will the hands of the continuously driven clock be opposite each other?
      • What time after 5:00 am will the hands of the continuously driven clock extend in opposite direction?
    • An integer is any whole number, including zero (0).
    • A fraction is a part or portion of something, usually expressed as a numerator over a denominator.
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