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1.2 – Solving Linear Equations

Objectives

  • Review steps for solving linear equations, including collecting like terms, using the order of operations, and performing inverse operations.
  • Identify equations that have no solutions or infinitely many solutions.
  • Ask the questions necessary to turn real-life problems into mathematical sentences.
  • Identify constraints that might affect the possible solutions in a real-life problem.

 

Key Terms

  • Distributive Property – the rule that if a, b, and c are numbers or expressions, then a (b + c) = a b + a c.
  • Infinitely – without end or limit; going on forever; impossible to count.
  • Inverse Operations – an inverse operation is an operation that reverses (undoes) the action of another operation.
  • Like Terms – terms in an algebraic expression that have the same variables raised to the same powers.
  • Linear Equation – any equation whose graph is a straight line. A linear equation can be written in the form y = mx + b.
  • Standard Form of a Linear Equation – a linear equation written in the form Ax + By + C = 0.

 

Notes

Distributive Property
  • Question: When do I need to use the distributive property?
    • Answer: When a variable is part of an expression inside parentheses that is multiplied by a number or variable, like 3(x + 4).
  • Question: Why do I need to use the distributive property?
    • Answer: So I can collect like terms and solve an equation.
Standard Form of a Linear Equation
  • a, b, and c are numbers & x is a variable
    • ax + b = c
    • ax – b = c
Solutions
  • No Solution – you will end up with an expression that is never true, no matter what the value of x is.
    • x = x + 7 (subtract x from both sides)
    • 0 = 7 (not true)

 

  • Infinitely Many Solutions – you will end up with an equation that is always true, no matter what the value of x is.
    • 3(x + 2) = 3x + 6 (distribute 3)
    • 3x + 6 = 3x + 6 (subtract 3x from both sides)
    • 6 = 6 (always true)

 

  • What is the solution?
    • Step 1: Write Equation => 2x + 2 = 10
    • Step 2: Isolate “x” using inverse operations, starting with constants => Subtract 2 on both sides (2x + 2 – 2 = 8 – 2)
    • Step 3: Isolate “x” using inverse operations, remove coefficients => divide by 2 on both sides:
      • \frac{2x}{2}\ = \frac{4}{2}
    • Step 4: Write your answer => x = 4

distribution

 

Examples

  • Jane is going to fence in her back yard. She has purchased 120 feet of fencing and knows that she wants to fence in a rectangular area where one side will be the back of her house. She knows her house is 25 feet across the back. Which of the following is the equation that Jane can use to figure out how far back from the house she can fence in?
  • Setup
    • x: how far back from Jane’s house
    • 3 sides of the fence: x (side), x (side), 25 (back of fence)
    • 4th side of yard is the back of Jane’s house (no fence needed)
    • x + x + 25 = 120
    • 2x + 25 = 120
    • 2x = 95
      • Answer:  x = 47.5 feet
  • Three friends all have ages that are consecutive integers. The sum of their ages is 180. What is the age of the oldest friend?
  • Setup
    • x: 1st friend (youngest)
    • x + 1: 2nd friend (one year older)
    • x + 2: 3rd friend (two years older, oldest)
    • x + X + 1 + x + 2 = 180
    • 3x + 3 = 180
    • 3x = 177
      • Answer:  x = 59, so 2nd friend = 60 and 3rd friend = 61 (oldest)

 

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