Simplifying Algebraic Expressions

Order of Operations (PEMDAS)

  • PEMDAS Framework: The foundational order of operations utilized to evaluate algebraic expressions:
    • P: Parentheses (and other grouping symbols)
    • E: Exponents
    • M: Multiplication
    • D: Division
    • A: Addition
    • S: Subtraction
  • Precedence Rules: Multiplication and division share equal precedence and are solved sequentially from left to right. Similarly, addition and subtraction share equal precedence and are solved sequentially from left to right.

Systematic Procedure for Evaluating Expressions

  • Step 1 (Substitution): Replace all variables in the expression with their designated numerical values. Enclose negative substituted values in parentheses to prevent sign errors.
  • Step 2 (Parentheses & Grouping Evaluation): Simplify terms located inside parentheses or within fractional numerators and denominators following standard operational hierarchy.
  • Step 3 (Exponents): Calculate the values of any terms containing exponents.
  • Step 4 (Multiplication and Division): Perform all multiplication and division operations in order from left to right.
  • Step 5 (Addition and Subtraction): Perform all remaining addition and subtraction operations in order from left to right to determine the final evaluated result.

Problem Sets and Complete Step-by-Step Solutions

  • Problem 1:

    • Expression: 7−14d+2h7 - \frac{14}{d} + 2h
    • Given Values: d=7d = 7, h=−6h = -6
    • Substitution: 7−147+2(−6)7 - \frac{14}{7} + 2(-6)
    • Step-by-Step Evaluation:
    • Evaluate the division term: 147=2\frac{14}{7} = 2
    • Evaluate the multiplication term: 2(−6)=−122(-6) = -12
    • Substitute back into the expression: 7−2+(−12)7 - 2 + (-12)
    • Perform subtraction and addition from left to right: 7−2=57 - 2 = 5, then 5+(−12)=−75 + (-12) = -7
    • Final Simplified Value: −7-7

  • Problem 2:

    • Expression: −3(−9−7c)−4w-3(-9 - 7c) - 4w
    • Given Values: c=6c = 6, w=7w = 7
    • Substitution: −3(−9−7(6))−4(7)-3(-9 - 7(6)) - 4(7)
    • Step-by-Step Evaluation:
    • Simplify inside parentheses: 7(6)=427(6) = 42, then −9−42=−51-9 - 42 = -51
    • Multiply outer factor with inner result: −3(−51)=153-3(-51) = 153
    • Evaluate the remaining multiplication: 4(7)=284(7) = 28
    • Subtract terms: 153−28=125153 - 28 = 125
    • Final Simplified Value: 125125

  • Problem 3:

    • Expression: z5+7+2h\frac{z}{5} + 7 + 2h
    • Given Values: z=−15z = -15, h=−9h = -9
    • Substitution: −155+7+2(−9)\frac{-15}{5} + 7 + 2(-9)
    • Step-by-Step Evaluation:
    • Evaluate the division term: −155=−3\frac{-15}{5} = -3
    • Evaluate the multiplication term: 2(−9)=−182(-9) = -18
    • Add and subtract from left to right: −3+7=4-3 + 7 = 4, then 4+(−18)=−144 + (-18) = -14
    • Final Simplified Value: −14-14

  • Problem 4:

    • Expression: 5(−8r+2d)+45(-8r + 2d) + 4
    • Given Values: r=6r = 6, d=−2d = -2
    • Substitution: 5(−8(6)+2(−2))+45(-8(6) + 2(-2)) + 4
    • Step-by-Step Evaluation:
    • Evaluate products inside parentheses: −8(6)=−48-8(6) = -48 and 2(−2)=−42(-2) = -4
    • Sum terms inside parentheses: −48+(−4)=−52-48 + (-4) = -52
    • Multiply outer factor: 5(−52)=−2605(-52) = -260
    • Add constant term: −260+4=−256-260 + 4 = -256
    • Final Simplified Value: −256-256

  • Problem 5:

    • Expression: 24n+8−2x\frac{24}{n} + 8 - 2x
    • Given Values: n=6n = 6, x=−4x = -4
    • Substitution: 246+8−2(−4)\frac{24}{6} + 8 - 2(-4)
    • Step-by-Step Evaluation:
    • Evaluate division: 246=4\frac{24}{6} = 4
    • Evaluate multiplication: −2(−4)=8-2(-4) = 8
    • Combine terms: 4+8+8=204 + 8 + 8 = 20
    • Final Simplified Value: 2020

  • Problem 6:

    • Expression: −5(3b−4)−6x-5(3b - 4) - 6x
    • Given Values: b=−7b = -7, x=−2x = -2
    • Substitution: −5(3(−7)−4)−6(−2)-5(3(-7) - 4) - 6(-2)
    • Step-by-Step Evaluation:
    • Simplify inside parentheses: 3(−7)=−213(-7) = -21, then −21−4=−25-21 - 4 = -25
    • Multiply outer factor: −5(−25)=125-5(-25) = 125
    • Evaluate remaining product: −6(−2)=12-6(-2) = 12
    • Combine terms: 125+12=137125 + 12 = 137
    • Final Simplified Value: 137137

  • Problem 7:

    • Expression: 2b−8(9h+5)2b - 8(9h + 5)
    • Given Values: h=−9h = -9, b=−2b = -2
    • Substitution: 2(−2)−8(9(−9)+5)2(-2) - 8(9(-9) + 5)
    • Step-by-Step Evaluation:
    • Simplify inside parentheses: 9(−9)=−819(-9) = -81, then −81+5=−76-81 + 5 = -76
    • Evaluate left product: 2(−2)=−42(-2) = -4
    • Evaluate right product: −8(−76)=608-8(-76) = 608
    • Combine terms: −4+608=604-4 + 608 = 604
    • Final Simplified Value: 604604

  • Problem 8:

    • Expression: −5+f4−9x-5 + \frac{f}{4} - 9x
    • Given Values: f=−36f = -36, x=4x = 4
    • Substitution: −5+−364−9(4)-5 + \frac{-36}{4} - 9(4)
    • Step-by-Step Evaluation:
    • Evaluate division: −364=−9\frac{-36}{4} = -9
    • Evaluate multiplication: 9(4)=369(4) = 36
    • Combine terms from left to right: −5+(−9)=−14-5 + (-9) = -14, then −14−36=−50-14 - 36 = -50
    • Final Simplified Value: −50-50

  • Problem 9:

    • Expression: 6h−7(−4+5n)6h - 7(-4 + 5n)
    • Given Values: n=8n = 8, h=−7h = -7
    • Substitution: 6(−7)−7(−4+5(8))6(-7) - 7(-4 + 5(8))
    • Step-by-Step Evaluation:
    • Simplify inside parentheses: 5(8)=405(8) = 40, then −4+40=36-4 + 40 = 36
    • Evaluate left multiplication: 6(−7)=−426(-7) = -42
    • Evaluate right multiplication: −7(36)=−252-7(36) = -252
    • Combine terms: −42−252=−294-42 - 252 = -294
    • Final Simplified Value: −294-294

  • Problem 10:

    • Expression: w3+9+4s\frac{w}{3} + 9 + 4s
    • Given Values: w=−6w = -6, s=9s = 9
    • Substitution: −63+9+4(9)\frac{-6}{3} + 9 + 4(9)
    • Step-by-Step Evaluation:
    • Evaluate division: −63=−2\frac{-6}{3} = -2
    • Evaluate multiplication: 4(9)=364(9) = 36
    • Combine terms from left to right: −2+9=7-2 + 9 = 7, then 7+36=437 + 36 = 43
    • Final Simplified Value: 4343

Visual Reference

Simplifying Algebraic Expressions Practice Sheet