Writing and Translating Equations

Defining Characteristics of an Equation

  • An equation is a mathematical statement asserting that two expressions possess the exact same quantitative value.

  • The essential defining characteristic of an equation is the equal sign ($=$).

  • An equal sign establishes a balance where the left side (LHS) of the statement has the exact same value as the right side (RHS).

  • Common verbal phrasing used to signify an equal sign ($=$) in mathematical word problems includes:

    • "is"

    • "equals"

    • "same as"

Translating Verbal Expressions into Equations

  • Translating written descriptions into mathematical equations requires converting verbal phrases into corresponding operations, variables, and relational symbols.

  • Translation Example 1:

    • Verbal statement: "8 times the difference of a number and 3 is the same as the square of a number plus 10."

    • Structural breakdown:

    • "8 times the difference of a number and 3" translates to: 8(x−3)8(x - 3)

    • "is the same as" translates to: ==

    • "the square of a number plus 10" translates to: x2+10x^2 + 10

    • Final translated equation: 8(x−3)=x2+108(x - 3) = x^2 + 10

  • Translation Example 2 (Geometric Formula):

    • Verbal statement: "The area of a triangle is half (or 12\frac{1}{2}) the product of its base and its height."

    • Variable identification:

    • Area is represented by AA

    • Base is represented by BB

    • Height is represented by HH

    • Formulated equations:

    • Standard fraction form: A=12BHA = \frac{1}{2}BH

    • Alternative division form: A=BH2A = \frac{BH}{2}

Practice Problems and Exercises

  • Homework Assignment Reference:

    • Pages: P. 71–73

  • Algebraic Equation Exercises:

    • Sample linear equation with grouping symbols:

    • 85−9y=7(4+y)85 - 9y = 7(4 + y)

    • Sample cost equation and application (Problem 8):

    • Linear cost function: C=39,000+3tC = 39,000 + 3t

    • Evaluated total cost equation: 216,000=39,000+3t216,000 = 39,000 + 3t

    • Consecutive term sequence expression (Problem 11):

    • x+x+1+x+2x + x + 1 + x + 2

    • Geometric perimeter formula (Problem 15):

    • Perimeter of a rectangle: P=2l+2wP = 2l + 2w