Algebraic Equations, Expansions, and Systems Study Guide

Algebraic Substitution and Formula Evaluation

  • Substituting numerical values into algebraic expressions requires replacing variables with given constants and following the correct order of operations (PEMDAS/BODMAS).

  • Evaluation Problem 1:

    • Formula: z=2x2+3yz = 2x^2 + 3y
    • Given values: x=−2x = -2, y=−5y = -5
    • Step 1 (Substitute variables): z=2(−2)2+3(−5)z = 2(-2)^2 + 3(-5)
    • Step 2 (Evaluate powers): (−2)2=4(-2)^2 = 4, so z=2(4)+3(−5)z = 2(4) + 3(-5)
    • Step 3 (Multiply): z=8−15z = 8 - 15
    • Step 4 (Simplify): z=−7z = -7

  • Evaluation Problem 2:

    • Formula: C=2x24+17yC = \frac{2x^2}{4} + \frac{17}{y}
    • Given values: x=−5x = -5, y=−3y = -3
    • Step 1 (Substitute variables): C=2(−5)24+17−3C = \frac{2(-5)^2}{4} + \frac{17}{-3}
    • Step 2 (Evaluate numerator of first term): 2(−5)2=2(25)=502(-5)^2 = 2(25) = 50
    • Step 3 (Simplify terms): 504=12.5\frac{50}{4} = 12.5 and 17−3=−5.67\frac{17}{-3} = -5.67
    • Step 4 (Combine terms): C=12.5−5.67=6.83C = 12.5 - 5.67 = 6.83

  • Evaluation Problem 3 (System of Proportional Relations):

    • Given relationships: ab=3c\frac{a}{b} = 3c and bc=2\frac{b}{c} = 2
    • Step 1: If ab=24\frac{a}{b} = 24, then b=4b = 4 when a=96a = 96
    • Step 2: From bc=2\frac{b}{c} = 2, if b=16b = 16, then c=8c = 8
    • Step 3: Substitute b=16b = 16 into a16=24\frac{a}{16} = 24 to obtain a=24×16=384a = 24 \times 16 = 384

  • Evaluation Problem 4:

    • Expression: 3w2−w33w^2 - w^3
    • Given value: w=−2w = -2
    • Step 1 (Substitute variable): 3(−2)2−(−2)33(-2)^2 - (-2)^3
    • Step 2 (Evaluate exponents): (−2)2=4(-2)^2 = 4 and (−2)3=−8(-2)^3 = -8
    • Step 3 (Perform operations): 3(4)−(−8)=12+8=203(4) - (-8) = 12 + 8 = 20

  • Evaluation Problem 5 (Kinematic Displacement Formula):

    • Formula: S=ut+12at2S = ut + \frac{1}{2}at^2
    • Given values: u=5u = 5, t=10t = 10, a=3a = 3
    • Step 1 (Substitute values): S=(5)(10)+12(3)(10)2S = (5)(10) + \frac{1}{2}(3)(10)^2
    • Step 2 (Evaluate first term): ut=50ut = 50
    • Step 3 (Evaluate second term): 12(3)(100)=0.5×300=150\frac{1}{2}(3)(100) = 0.5 \times 300 = 150
    • Step 4 (Sum terms): S=50+150=200S = 50 + 150 = 200

Expansion and Simplification of Algebraic Expressions

  • Polynomial expansion relies on distributing every term in the first factor to every term in the second factor.

  • Binomial Expansion 1:

    • Expression: (x+3)(x+5)(x + 3)(x + 5)
    • Step 1 (Distribute terms): x2+5x+3x+15x^2 + 5x + 3x + 15
    • Step 2 (Combine like terms): x2+8x+15x^2 + 8x + 15

  • Binomial Expansion 2:

    • Expression: (3x−7)(2x+9)(3x - 7)(2x + 9)
    • Step 1 (Distribute terms): 6x2+27x−14x−636x^2 + 27x - 14x - 63
    • Step 2 (Combine like terms): 6x2+13x−636x^2 + 13x - 63

  • Monomial-Binomial Distribution:

    • Expression: 2x(3−x2)2x(3 - x^2)
    • Distribution step: 2x(3)−2x(x2)=6x−2x32x(3) - 2x(x^2) = 6x - 2x^3

  • Binomial Expansion 3:

    • Expression: (2p+3)(3p−2)(2p + 3)(3p - 2)
    • Step 1 (Distribute terms): 6p2−4p+9p−66p^2 - 4p + 9p - 6
    • Step 2 (Combine like terms): 6p2+5p−66p^2 + 5p - 6

  • Binomial Expansion 4:

    • Expression: (x+2)(2x+1)(x + 2)(2x + 1)
    • Step 1 (Distribute terms): 2x2+x+4x+22x^2 + x + 4x + 2
    • Step 2 (Combine like terms): 2x2+5x+22x^2 + 5x + 2

  • Like-Term Simplification Examples:

    • Example A: 4c2+6c+2c+3=4c2+8c+34c^2 + 6c + 2c + 3 = 4c^2 + 8c + 3
    • Example B: 10x2+27x+510x^2 + 27x + 5

Solving One-Step and Two-Step Linear Equations

  • Solving linear equations involves using inverse operations to isolate the unknown variable xx on one side of the equality sign.

  • Exercise 1 Solutions:

    • Part a:
    • Equation: x+9=4x + 9 = 4
    • Operation: Subtract 99 from both sides
    • Solution: x=−5x = -5
    • Part b:
    • Equation: 5x=455x = 45
    • Operation: Divide both sides by 55
    • Solution: x=9x = 9
    • Part c:
    • Equation: −24=−6x-24 = -6x
    • Operation: Divide both sides by −6-6
    • Solution: x=4x = 4
    • Part d:
    • Equation: 3−x=123 - x = 12
    • Step 1: Subtract 33 from both sides to get −x=9-x = 9
    • Step 2: Multiply by −1-1
    • Solution: x=−9x = -9
    • Part e:
    • Equation: 2x+5=172x + 5 = 17
    • Step 1: Subtract 55 from both sides to get 2x=122x = 12
    • Step 2: Divide by 22
    • Solution: x=6x = 6
    • Part f:
    • Equation: 3x−2=−143x - 2 = -14
    • Step 1: Add 22 to both sides to get 3x=−123x = -12
    • Step 2: Divide by 33
    • Solution: x=−4x = -4
    • Part g:
    • Equation: 3−4x=−173 - 4x = -17
    • Step 1: Subtract 33 from both sides to get −4x=−20-4x = -20
    • Step 2: Divide by −4-4
    • Solution: x=5x = 5
    • Part h:
    • Equation: 8=9−2x8 = 9 - 2x
    • Step 1: Subtract 99 from both sides to get −1=−2x-1 = -2x
    • Step 2: Divide by −2-2
    • Solution: x=0.5x = 0.5

Solving Equations with Parentheses and Variables on Both Sides

  • Equations containing brackets must first be expanded using the distributive property, combined across like terms, and then rearranged so all terms with variables are on one side and constant values are on the other.

  • Exercise 2 Solutions (Equations with Brackets):

    • Part 1a:
    • Equation: 2(x+8)+5(x−1)=602(x + 8) + 5(x - 1) = 60
    • Expand brackets: 2x+16+5x−5=602x + 16 + 5x - 5 = 60
    • Combine like terms: 7x+11=607x + 11 = 60
    • Subtract 1111: 7x=497x = 49
    • Divide by 77: x=7x = 7
    • Part 1b:
    • Equation: 2(x−3)+3(x+2)=−52(x - 3) + 3(x + 2) = -5
    • Expand brackets: 2x−6+3x+6=−52x - 6 + 3x + 6 = -5
    • Combine like terms: 5x=−55x = -5
    • Divide by 55: x=−1x = -1
    • Part 1c:
    • Equation: 3(x+3)−2(x+1)=03(x + 3) - 2(x + 1) = 0
    • Expand brackets: 3x+9−2x−2=03x + 9 - 2x - 2 = 0
    • Combine like terms: x+7=0x + 7 = 0
    • Subtract 77: x=−7x = -7
    • Part 1d:
    • Equation: 4(2x−3)+2(x+2)=324(2x - 3) + 2(x + 2) = 32
    • Expand brackets: 8x−12+2x+4=328x - 12 + 2x + 4 = 32
    • Combine like terms: 10x−8=3210x - 8 = 32
    • Add 88: 10x=4010x = 40
    • Divide by 1010: x=4x = 4
    • Part 1e:
    • Equation: 3(4x+1)−2(3x−4)=−73(4x + 1) - 2(3x - 4) = -7
    • Expand brackets: 12x+3−6x+8=−712x + 3 - 6x + 8 = -7
    • Combine like terms: 6x+11=−76x + 11 = -7
    • Subtract 1111: 6x=−186x = -18
    • Divide by 66: x=−3x = -3
    • Part 1f:
    • Equation: 5(x+2)−2(3−2x)=−145(x + 2) - 2(3 - 2x) = -14
    • Expand brackets: 5x+10−6+4x=−145x + 10 - 6 + 4x = -14
    • Combine like terms: 9x+4=−149x + 4 = -14
    • Subtract 44: 9x=−189x = -18
    • Divide by 99: x=−2x = -2

  • Equations with Variables on Both Sides:

    • Part 2a:
    • Equation: 2x−3=3x+62x - 3 = 3x + 6
    • Subtract 2x2x from both sides: −3=x+6-3 = x + 6
    • Subtract 66 from both sides: x=−9x = -9
    • Part 2b:
    • Equation: 3x−4=−5−x3x - 4 = -5 - x
    • Add xx to both sides: 4x−4=−54x - 4 = -5
    • Add 44 to both sides: 4x=−14x = -1
    • Divide by 44: x=−14x = -\frac{1}{4}
    • Part 2c:
    • Equation: 4−5x=3x+64 - 5x = 3x + 6
    • Add 5x5x and subtract 66: −2=8x-2 = 8x
    • Divide by 88: x=−28=−14x = -\frac{2}{8} = -\frac{1}{4}
    • Part 2d:
    • Equation: −x=2x+4-x = 2x + 4
    • Subtract 2x2x from both sides: −3x=4-3x = 4
    • Divide by −3-3: x=−43=−113x = -\frac{4}{3} = -1\frac{1}{3}
    • Part 2e:
    • Equation: 12−7x=3x+712 - 7x = 3x + 7
    • Add 7x7x and subtract 77: 5=10x5 = 10x
    • Divide by 1010: x=510=12x = \frac{5}{10} = \frac{1}{2}
    • Part 2f:
    • Equation: 5x−9=1−3x5x - 9 = 1 - 3x
    • Add 3x3x and add 99: 8x=108x = 10
    • Divide by 88: x=108=128=114x = \frac{10}{8} = 1\frac{2}{8} = 1\frac{1}{4}
    • Part 2g:
    • Equation: 4−x−2(2−x)=6x4 - x - 2(2 - x) = 6x
    • Expand brackets: 4−x−4+2x=6x4 - x - 4 + 2x = 6x
    • Simplify left side: x=6xx = 6x
    • Subtract xx: 0=5x  ⟹  x=00 = 5x \implies x = 0
    • Part 2h:
    • Equation: 5−3(1−x)=2−3x5 - 3(1 - x) = 2 - 3x
    • Expand brackets: 5−3+3x=2−3x5 - 3 + 3x = 2 - 3x
    • Simplify left side: 2+3x=2−3x2 + 3x = 2 - 3x
    • Add 3x3x and subtract 22: 6x=0  ⟹  x=06x = 0 \implies x = 0
    • Part 2i:
    • Equation: 5−2x−(2x+1)=−65 - 2x - (2x + 1) = -6
    • Expand brackets: 5−2x−2x−1=−65 - 2x - 2x - 1 = -6
    • Combine like terms: 4−4x=−64 - 4x = -6
    • Subtract 44: −4x=−10-4x = -10
    • Divide by −4-4: x=−10−4=52=2.5x = \frac{-10}{-4} = \frac{5}{2} = 2.5

Solving Equations with Fractions and Denominators

  • Fractional equations are solved by multiplying the entire equation by the least common multiple (LCM) of the denominators to clear all fractions.

  • Simple Fractional Equations:

    • Part 2a: x4=12  ⟹  x=12×4=48\frac{x}{4} = 12 \implies x = 12 \times 4 = 48
    • Part 2b: 12x=6  ⟹  x=60.5=12\frac{1}{2}x = 6 \implies x = \frac{6}{0.5} = 12
    • Part 2c: 5=x−2  ⟹  x=5×(−2)=−105 = \frac{x}{-2} \implies x = 5 \times (-2) = -10
    • Part 2d: x3+4=−2  ⟹  x3=−6  ⟹  x=−18\frac{x}{3} + 4 = -2 \implies \frac{x}{3} = -6 \implies x = -18
    • Part 2e: x+35=−2  ⟹  x+3=−10  ⟹  x=−13\frac{x+3}{5} = -2 \implies x + 3 = -10 \implies x = -13
    • Part 2f: 13(x+2)=3  ⟹  x+2=9  ⟹  x=7\frac{1}{3}(x + 2) = 3 \implies x + 2 = 9 \implies x = 7
    • Part 2g: 2x−13=7  ⟹  2x−1=21  ⟹  2x=22  ⟹  x=11\frac{2x - 1}{3} = 7 \implies 2x - 1 = 21 \implies 2x = 22 \implies x = 11
    • Part 2h: 12(5−x)=−2  ⟹  2.5−0.5x=−2  ⟹  −0.5x=−4.5  ⟹  x=9\frac{1}{2}(5 - x) = -2 \implies 2.5 - 0.5x = -2 \implies -0.5x = -4.5 \implies x = 9

  • Multi-Step Fractional Equations (Exercise 3):

    • Problem 1d:
    • Equation: x+13=2x−14\frac{x+1}{3} = \frac{2x-1}{4}
    • Clear denominators by multiplying both sides by the LCM, 1212
    • Step 1: 4(x+1)=3(2x−1)4(x + 1) = 3(2x - 1)
    • Step 2 (Expand): 4x+4=6x−34x + 4 = 6x - 3
    • Step 3 (Rearrange): 4+3=6x−4x4 + 3 = 6x - 4x
    • Step 4 (Solve): 7=2x  ⟹  x=727 = 2x \implies x = \frac{7}{2}
    • Problem 2d:
    • Equation: 32x=74\frac{3}{2x} = \frac{7}{4}
    • Cross-multiply: 6×3=14x6 \times 3 = 14x
    • Step 1: 18=14x18 = 14x
    • Step 2: x=1814=97x = \frac{18}{14} = \frac{9}{7}
    • Problem 2h:
    • Equation: 47x=32x\frac{4}{7x} = \frac{3}{2x}
    • Cross-multiply: 4(2x)=3(7x)  ⟹  8x=21x4(2x) = 3(7x) \implies 8x = 21x
    • Step 1: 21x−8x=0  ⟹  13x=021x - 8x = 0 \implies 13x = 0
    • Result: x=0x = 0 (Since xx is in the denominator, x≠0x \neq 0, so no valid non-zero solution exists).

Systems of Simultaneous Linear Equations

  • Simultaneous equations involve finding values of two variables that satisfy two linear equations at the same time, using elimination or substitution methods.

  • System 1:

    • Given system:
    • Equation 1: x+y=7x + y = 7
    • Equation 2: 2x+y=122x + y = 12
    • Elimination Method (Subtract Equation 1 from Equation 2):
    • (2x+y)−(x+y)=12−7(2x + y) - (x + y) = 12 - 7
    • x=5x = 5
    • Substitute x=5x = 5 into Equation 1:
    • 5+y=7  ⟹  y=25 + y = 7 \implies y = 2
    • Alternative linear transformation worked out in exercises:
    • Given 4x+2y=104x + 2y = 10 and x+2y=7x + 2y = 7
    • Subtract equations: 3x=3  ⟹  x=13x = 3 \implies x = 1
    • Substitute x=1x = 1 into 4(1)+2y=10  ⟹  2y=6  ⟹  y=34(1) + 2y = 10 \implies 2y = 6 \implies y = 3

  • System 2:

    • Given system:
    • Equation 1: 4x+2y=344x + 2y = 34
    • Equation 2: 3x+y=213x + y = 21
    • Scale equations to create equal coefficients for xx:
    • Multiply Equation 1 by 33: 12x+6y=10212x + 6y = 102
    • Multiply Equation 2 by 44: 12x+4y=8412x + 4y = 84
    • Subtract scaled equations:
    • (12x+6y)−(12x+4y)=102−84(12x + 6y) - (12x + 4y) = 102 - 84
    • 2y=18  ⟹  y=92y = 18 \implies y = 9
    • Substitute y=9y = 9 into Equation 2:
    • 3x+9=21  ⟹  3x=12  ⟹  x=43x + 9 = 21 \implies 3x = 12 \implies x = 4

  • System 3:

    • Given system:
    • Equation 1: 6x+y=186x + y = 18
    • Equation 2: 4x+y=144x + y = 14
    • Scale factors used in work: Multiply Equation 1 by 44 and Equation 2 by 66
    • Equation 1 scaled: 24x+4y=7224x + 4y = 72
    • Equation 2 scaled: 24x+6y=8424x + 6y = 84
    • Direct elimination of yy by subtracting Equation 2 from Equation 1:
    • (6x+y)−(4x+y)=18−14(6x + y) - (4x + y) = 18 - 14
    • 2x=4  ⟹  x=22x = 4 \implies x = 2
    • Substitute x=2x = 2 into Equation 1:
    • 6(2)+y=18  ⟹  12+y=18  ⟹  y=66(2) + y = 18 \implies 12 + y = 18 \implies y = 6