Solving Linear Equations

Solving Linear Equations

Problem 46

The equation provided is: −8z+2z=4+14-8z + 2z = 4 + 14

  • Step 1: Combine like terms on both sides.
    • On the left side: −8z+2z=−6z-8z + 2z = -6z
    • On the right side: 4+14=184 + 14 = 18
    • The equation simplifies to: −6z=18-6z = 18
  • Step 2: Isolate the variable zz by dividing both sides by -6.
    • −6z−6=18−6\frac{-6z}{-6} = \frac{18}{-6}
    • z=−3z = -3

Problem 47

The equation provided is: w−13=56w - \frac{1}{3} = \frac{5}{6}

  • Step 1: Isolate the variable ww by adding 13\frac{1}{3} to both sides.
    • w−13+13=56+13w - \frac{1}{3} + \frac{1}{3} = \frac{5}{6} + \frac{1}{3}
  • Step 2: Find a common denominator to add the fractions on the right side. The least common denominator for 6 and 3 is 6.
    • 13=1×23×2=26\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}
  • Step 3: Add the fractions.
    • w=56+26w = \frac{5}{6} + \frac{2}{6}
    • w=76w = \frac{7}{6}

Problem 48

The equation provided is: 28=6z−4z−3428 = 6z - 4z - 34

  • Step 1: Combine like terms on the right side.
    • 6z−4z=2z6z - 4z = 2z
    • The equation simplifies to: 28=2z−3428 = 2z - 34
  • Step 2: Isolate the term with zz by adding 34 to both sides.
    • 28+34=2z−34+3428 + 34 = 2z - 34 + 34
    • 62=2z62 = 2z
  • Step 3: Isolate the variable zz by dividing both sides by 2.
    • 622=2z2\frac{62}{2} = \frac{2z}{2}
    • z=31z = 31

Problem 49

The equation provided is: −12w−16=2(−6w−6)-12w - 16 = 2(-6w - 6)

  • Step 1: Distribute the 2 on the right side of the equation.
    • 2(−6w−6)=−12w−122(-6w - 6) = -12w - 12
    • The equation becomes: −12w−16=−12w−12-12w - 16 = -12w - 12
  • Step 2: Add 12w12w to both sides of the equation to eliminate the ww terms.
    • −12w−16+12w=−12w−12+12w-12w - 16 + 12w = -12w - 12 + 12w
    • −16=−12-16 = -12
  • Step 3: Analyze the result.
    • Since −16=−12-16 = -12 is not a true statement, there is no solution to this equation.
  • Conclusion:
    • The answer is B) No Solution