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1.9 – Solving Multistep Linear Equations

Objectives

  • Isolate the variables in an equation or inequality by identifying the operations involved.
  • Collect like terms to simplify the equation or inequality.
  • Remember the order of operations for solving an equation or inequality in one variable in the proper sequence.
  • Identify equations that have no solution or an infinite number of solutions.
  • Ask the questions necessary to turn real-life problems into mathematical sentences and equations in one variable.

 

Key Terms

  • Distributive Property – The rule that if a, b, and c are numbers or expressions, then a(b + c) = ab + ac.
    • Example
      • Step 1: Start with the original equation
        • 3(x + 4) – 2x = 5
      • Step 2: Distribute (using multiplication)
        • 3x + 12 – 2x = 5
      • Step 3: Collect like terms
        • (3x – 2x) + 12 = 5
      • Step 4: Simplify
        • x + 12 = 5
      • Step 5: Solve
        • X = -7

 

  • Infinite – Without end or limit; going on forever; impossible to count.
  • Like Terms – Terms in an algebraic expression that have the same variables raised to the same powers.
    • 5ab and -22ab are like terms because they have the same variables in each term.
    • 8y and -5x are not like terms because they have different variables.

Alg1A 1.09 - Terms

  • Linear Equations – Equations whose graphs are straight lines. A linear equation can be written in the form y = mx + b.

 

Notes

  • To solve these types of equations, you have to “undo” the variable expression. First you have to isolate the variable term, and then you have to isolate the variable itself.
    • To undo (cancel) addition (positives), you must subtract terms on BOTH sides.
    • To undo (cancel) subtraction (negatives), you must add the terms to BOTH sides.
    • To undo (cancel) multiplication, you must divide terms or coefficients on BOTH sides.
    • To undo (cancel) division, you must multiply terms or coefficients on BOTH sides.

Alg1A 1.09 - Rules for Undoing

Alg1A 1.09 - Multistep Inequality

  • No Solution
    • When an equation has no solution, there is no value for x that would make the equation true.
    • Example: for x = x + 7, no matter what value you use for x, you will always get the untrue statement 0 = 7.
      • Start with the equation: x = x + 7
      • Subtract x on both sides: 0 = 7
      • 7 does not equal zero!  So, this is untrue; therefore, no solution!

 

  • Infinite Solutions
    • If you try to solve an equation that has an infinite number of solutions, you will end up with an equation that is always true, no matter what the value of x is.
    • Example: if x = x is the answer (or if two sides of the equation are identical), every solution is possible; so, infinite solutions is the answer.
      • Start with the equation: 3 (x + 2) = 3x + 6
      • Distribute 3 into the parenthesis: 3x + 6 = 3x + 6
        • Notice how the two sides are identical!
      • Subtract 6 from both sides: 3x = 3x
      • Divide by 3 on both sides: x = x
      • So, for ANY value you substitute into x, both sides will ALWAYS be equal.

 


  • Example: 3x + 4 = 13
    • Step 1: Subtract 4 from both sides to isolate the variable term
      • 3x = 9
    • Step 2: Divide both sides by 3 to isolate the variable
      • x = 3

 


  • Example: 3x – 4 = 17
    • Step 1: Add 4 to both sides to isolate the variable term
      • 3x = 21
    • Step 2: Divide both sides by 3 to isolate the variable
      • x = 7

 


  • Example
    • Step 1: Start with the equation
      • \frac{1}{2}(4x + 4) = x +\frac{1}{2}(4 - 6x) + 8
    • Step 2: Distribute (multiply 1/2 into each term inside the parenthesis)
      • 2x + 2 = x + 2 - 3x + 8
    • Step 3: Collect like terms
      • 2x + 2 = (x - 3x) + (2 + 8)
    • Step 4: Simplify (combine like terms)
      • 2x + 2 = -2x + 10
    • Step 5: Add 2x to both sides
      • 4x + 2 = 10
    • Step 6: Subtract 2 from both sides
      • 4x = 8
    • Step 7: Divide both sides by 4 and write the answer
      • x = 2

 

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