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1.10 – Solving Literal Equations and Formulas

Objectives

  • Solve literal equations with two variables.
  • Rearrange formulas to solve for a quantity of interest in a real-world problem.
  • Describe the steps in solving equations.
  • Use units to guide and interpret your solutions of literal equations and formulas.

 

Key Terms

  • Fahrenheit and Celsius – Ways to calculate and tell the temperature.
    • In the USA, we use Fahrenheit. In Europe (and many other countries), they use Celsius.
    • Formula for finding Celsius:
      • C=\frac{5}{9}(F-32)
    • What if you wanted to solve for Fahrenheit instead?
      • Try solving for F by doing inverse (reverse) operations until you isolate the F.
      • Step 1: Multiply by 9 on both sides of the equation
        • 9C=9*\frac{5}{9}(F-32) and this will give you: 9C=5(F-32)
      • Step 2: Divide by 5 on both sides of the equation
        • \frac{9}{5}C=5\frac{(F-32)}{5} and this will give you: \frac{9}{5}C=F-32
      • Step 3: Add 32 to both sides of the equation
        • You get the formula for finding Fahrenheit: \frac{9}{5}C+32=F
        • You can rewrite it as F=\frac{9}{5}C+32
  • Formula – An equation that describes an important mathematical relationship.
  • Literal Equation – An equation that involves two or more variables.
  • Ohm’s Law  – Describes the relationships among voltage, current, and resistance in a conductor (ex. computer circuits)
    • The voltage of the current is V.
    • The amount of current flowing through the conductor is I
    • Any resistance that is present is measured with R
    • Formula: V = IR
    • Example: If a voltage of 0.5 V (volts) is driving a current of 0.03 A (amps) through a resistor, what is the resistance fo the resistor, measured in ohms?
      • The symbol for ohms is Omega: \Omega
      • Step 1: List what we know and the formula
        • V: 0.5
        • I: 0.03
        • Formula: V = IR
        • Variable (unknown): R
      • Step 2: Solve for R by dividing by I on both sides to isolate R
        • \frac{V}{I} = \frac{IR}{I}
        • \frac{V}{I} = R
      • Step 3: Substitute and solve
        • \frac{0.5}{0.03} = R
        • 16.667 \Omega = R
        • The resistance of a computer circuit is 16.667 ohms
  • Perimeter – The distance around an object
    • Perimeter of a rectangle (formula): P = 2l + 2w
      • P = perimeter
      • l = length
      • w = width
    • Perimeter Example: Felicia has 60 feet of fencing. She’d like to use all of this fencing to enclose a rectangular space in her backyard for a vegetable garden. The length of the garden must be 10 feet. What will the width be?
      • Step 1: List the facts & formula
        • P = 2l + 2w
        • Perimeter (fencing): 60ft
        • Space: rectangular
        • length: 10ft
        • width: ?
      • Step 2: Solve for w
        • Alg 1A - 1.10 - Permiter for w
      • Step 3: Substitute known variables
        • \frac{60}{2} - 10 = w
      • Step 4: Simplify and solve
        • 30 - 10 = w
        • w = 20
        • The width of the garden is 20 feet long.

 


 

Notes

  • Literal Equation Example 1: Solve for y
    • Step 1: Write the problem
      • 20x + 5y = 40
    • Step 2: Subtract 20x on both sides
      • 5y = 40 - 20x
    • Step 3: Divide both sides by 5
      • \frac{5y}{5} = \frac{40 - 20x}{5}
    • Step 4: Separate the fraction on the right side into two terms
      • \frac{5y}{5} = \frac{40}{5} - \frac{20x}{5}
    • Step 5: Simplify
      • y = 8 - 4x

 

  • Literal Equation Example 2: Solve for x
    • Step 1: Write the problem
      • 20x + 5y = 40
    • Step 2: Subtract 5y on both sides
      • 20x = 40 - 5y
    • Step 3: Divide both sides by 20
      • \frac{20x}{20} = \frac{40 - 5y}{20}
    • Step 4: Separate the fraction on the right side into two terms
      • \frac{20x}{20} = \frac{40}{20} - \frac{5y}{20}
    • Step 5: Simplify
      • x = 2 - \frac{y}{4}

 

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