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Solve the equation. Check your solution. (1)/(4)(-16 k+52)=8+10 k The solution set is _____ (Type an integer or a simplified fraction).

Solve the equation. Check your solution.\newline14(−16k+52)=8+10k\frac{1}{4}(-16 k+52)=8+10 k\newlineThe solution set is _____\_\_\_\_\_ (Type an integer or a simplified fraction).

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Full solution

Q. Solve the equation. Check your solution.\newline14(−16k+52)=8+10k\frac{1}{4}(-16 k+52)=8+10 k\newlineThe solution set is _____\_\_\_\_\_ (Type an integer or a simplified fraction).
  1. Distribute and Simplify: First, we need to simplify the equation by distributing the (1/4)(1/4) across the terms inside the parentheses.\newlineSo, (1/4)(−16k)+(1/4)(52)=8+10k(1/4)(-16k) + (1/4)(52) = 8 + 10k.\newlineThis simplifies to −4k+13=8+10k-4k + 13 = 8 + 10k.
  2. Move Terms: Next, we will move all terms containing kk to one side of the equation and the constant terms to the other side.\newlineTo do this, we add 4k4k to both sides and subtract 88 from both sides.\newline−4k+4k+13−8=8−8+10k+4k-4k + 4k + 13 - 8 = 8 - 8 + 10k + 4k.\newlineThis simplifies to 5=14k5 = 14k.
  3. Solve for kk: Now, we will solve for kk by dividing both sides of the equation by 1414.514=14k14\frac{5}{14} = \frac{14k}{14}.This simplifies to k=514k = \frac{5}{14}.
  4. Check Solution: Finally, we check our solution by substituting k=514k = \frac{5}{14} back into the original equation.(14)(−16(514)+52)=8+10(514).(\frac{1}{4})(-16(\frac{5}{14}) + 52) = 8 + 10(\frac{5}{14}).Simplifying the left side: (14)(−8014+52)=8+5014.(\frac{1}{4})(-\frac{80}{14} + 52) = 8 + \frac{50}{14}.Further simplifying: (14)(−8014+72814)=8+5014.(\frac{1}{4})(-\frac{80}{14} + \frac{728}{14}) = 8 + \frac{50}{14}.Combining like terms: (14)(64814)=8+5014.(\frac{1}{4})(\frac{648}{14}) = 8 + \frac{50}{14}.Simplifying: 16214=8+5014.\frac{162}{14} = 8 + \frac{50}{14}.Converting 88 to a fraction with a denominator of 1414: 16214=11214+5014.\frac{162}{14} = \frac{112}{14} + \frac{50}{14}.Adding the fractions on the right side: 16214=16214.\frac{162}{14} = \frac{162}{14}.Since both sides are equal, our solution checks out.

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