Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

(y-k)y=(1)/(3) In the given equation, k is a constant. One of the solutions to the equation is: (3+sqrt(9+4((1)/(3))))/(2) What is the value of k ?

(y−k)y=13 (y-k) y=\frac{1}{3} \newlineIn the given equation, k k is a constant. One of the solutions to the equation is:\newline3+9+4(13)2 \frac{3+\sqrt{9+4\left(\frac{1}{3}\right)}}{2} \newlineWhat is the value of k k ?

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 1right chevron icon
Domain and range of square root functions: equationsright chevron icon

Full solution

Q. (y−k)y=13 (y-k) y=\frac{1}{3} \newlineIn the given equation, k k is a constant. One of the solutions to the equation is:\newline3+9+4(13)2 \frac{3+\sqrt{9+4\left(\frac{1}{3}\right)}}{2} \newlineWhat is the value of k k ?
  1. Given equation and solution: We are given the equation (y−k)y=13(y-k)y = \frac{1}{3} and a solution y=3+9+4(13)2y = \frac{3+\sqrt{9+4(\frac{1}{3})}}{2}. To find the value of kk, we will substitute the given solution into the original equation and solve for kk.
  2. Substituting the solution: First, let's substitute y=3+9+4(13)2y = \frac{3+\sqrt{9+4\left(\frac{1}{3}\right)}}{2} into the equation (y−k)y=13(y-k)y = \frac{1}{3}. \newline(3+9+4(13)2−k)⋅3+9+4(13)2=13\left(\frac{3+\sqrt{9+4\left(\frac{1}{3}\right)}}{2} - k\right) \cdot \frac{3+\sqrt{9+4\left(\frac{1}{3}\right)}}{2} = \frac{1}{3}
  3. Simplifying the square root term: Now, let's simplify the square root term in the solution. \newlineWe have 9+4(13)=9+43\sqrt{9+4(\frac{1}{3})} = \sqrt{9+\frac{4}{3}} \newline=9+1.3333= \sqrt{9+1.3333} \newline=10.3333= \sqrt{10.3333}
  4. Substituting the simplified square root: Substitute the simplified square root back into the equation: \newline(3+10.33332−k)⋅3+10.33332=13(\frac{3+\sqrt{10.3333}}{2} - k) \cdot \frac{3+\sqrt{10.3333}}{2} = \frac{1}{3} \newline(3+10.33332)2−k⋅3+10.33332=13(\frac{3+\sqrt{10.3333}}{2})^{2} - k \cdot \frac{3+\sqrt{10.3333}}{2} = \frac{1}{3}
  5. Squaring the solution: Now, let's square the solution yy to prepare for substitution: \newline(3+10.33332)2=9+3⋅10.3333+3⋅10.3333+10.33334=19.3333+6⋅10.33334\left(\frac{3+\sqrt{10.3333}}{2}\right)^2 = \frac{9 + 3\cdot\sqrt{10.3333} + 3\cdot\sqrt{10.3333} + 10.3333}{4} = \frac{19.3333 + 6\cdot\sqrt{10.3333}}{4}
  6. Substituting y2y^2 back into the equation: Substitute y2y^2 back into the equation:\newline19.3333+6⋅10.33334−k⋅3+10.33332=13\frac{19.3333 + 6\cdot\sqrt{10.3333}}{4} - k \cdot \frac{3+\sqrt{10.3333}}{2} = \frac{1}{3}
  7. Clearing the fraction: Multiply both sides of the equation by 33 to clear the fraction: \newline3×(19.3333+610.33334−k⋅3+10.33332)=13 \times \left(\frac{19.3333 + 6\sqrt{10.3333}}{4} - k \cdot \frac{3+\sqrt{10.3333}}{2}\right) = 1
  8. Simplifying the left side: Simplify the left side of the equation:\newline(19.3333+610.33334×3−3k⋅3+10.33332)=1\left(\frac{19.3333 + 6\sqrt{10.3333}}{4} \times 3 - 3k \cdot \frac{3+\sqrt{10.3333}}{2}\right) = 1
  9. Distributing the 33: Now, let's distribute the 33 on the left side of the equation:\newline 58+1810.33334−9k+3k10.33332=1\frac{58 + 18\sqrt{10.3333}}{4} - \frac{9k+3k\sqrt{10.3333}}{2} = 1
  10. Having a common denominator: To simplify further, we need to have a common denominator for the terms on the left side. The common denominator is 44, so we multiply the second term by 22\frac{2}{2} to get the same denominator:\newline58+1810.33334−18k+6k10.33334=1\frac{58 + 18\sqrt{10.3333}}{4} - \frac{18k + 6k\sqrt{10.3333}}{4} = 1
  11. Combining like terms: Combine like terms on the left side:\newline(58+1810.3333−18k−6k10.33334)=1(\frac{58 + 18\sqrt{10.3333} - 18k - 6k\sqrt{10.3333}}{4}) = 1
  12. Clearing the denominator: Multiply both sides by 44 to clear the denominator: \newline58+1810.3333−18k−6k10.3333=458 + 18\sqrt{10.3333} - 18k - 6k\sqrt{10.3333} = 4
  13. Isolating k: Rearrange the terms to isolate k: \newline18k+6k10.3333=58+1810.3333−418k + 6k\sqrt{10.3333} = 58 + 18\sqrt{10.3333} - 4
  14. Simplifying the right side: Simplify the right side: \newline18k+6k10.3333=54+1810.333318k + 6k\sqrt{10.3333} = 54 + 18\sqrt{10.3333}
  15. Factoring out kk: Factor out kk on the left side:\newlinek(18+610.3333)=54+1810.3333k(18 + 6\sqrt{10.3333}) = 54 + 18\sqrt{10.3333}
  16. Dividing both sides by (18+610.3333)(18 + 6\sqrt{10.3333}): Divide both sides by (18+610.3333)(18 + 6\sqrt{10.3333}) to solve for kk:k=54+1810.333318+610.3333k = \frac{54 + 18\sqrt{10.3333}}{18 + 6\sqrt{10.3333}}
  17. Simplifying the right side: Simplify the right side by dividing both the numerator and the denominator by 66:\newlinek=6(9+310.3333)6(3+10.3333)k = \frac{6(9 + 3\sqrt{10.3333})}{6(3 + \sqrt{10.3333})} \newlinek=9+310.33333+10.3333k = \frac{9 + 3\sqrt{10.3333}}{3 + \sqrt{10.3333}}
  18. Solve for the value of kk: \newlinek=3(3+10.3333)3+10.3333k = \frac{3(3 + \sqrt{10.3333})}{3 + \sqrt{10.3333}}\newlinek=3k = 3

More problems from Domain and range of square root functions: equations

Question
What is the domain of this function? \newliney=xy = \sqrt{x} \newlineChoices: \newline{x∣x≥0}\{x|x \geq 0\}\newline{x∣x≤0}\{x|x \leq 0\}\newline{x∣x<0}\{x | x < 0\}\newline{x∣x>0}\{x|x>0\}
Get tutor helpright-arrow

Posted 9 months ago

Question
Solve for p p .\newline8=p 8 = \sqrt{p} \newlinep= p = _____
Get tutor helpright-arrow

Posted 9 months ago

Question
Solve for p p .\newline4p=10+3p \sqrt{4p} = \sqrt{10 + 3p} \newlinep= p = _____
Get tutor helpright-arrow

Posted 5 months ago

Question
What is the domain of this function? \newliney=xy = \sqrt{x}
Get tutor helpright-arrow

Posted 5 months ago

Question
Solve for p p .\newline8=p 8 = \sqrt{p} \newlinep= p = _____
Get tutor helpright-arrow

Posted 9 months ago

Question
Find the derivative of p(x) p(x) . \newlinep(x)=ln⁡(x) p(x) = \ln(\sqrt{x}) \newlinep′(x)= p' (x) = ______
Get tutor helpright-arrow

Posted 9 months ago

Question
Alice studies the relationship between climate and heart disease around the world.\newlineH(t) H(t) models the probability for the occurrence of heart disease (in percents relative to the global average) at an area where the temperature is t t degrees Celsius.\newlineAccording to Alice's model, when the temperature is −5∘C -5^{\circ}C (which is the lowest temperature included in the model), the probability is 10% 10\% above average. Then the probability decreases until the temperature reaches 30∘C 30^{\circ}C (which is the highest temperature included in the model), where the probability is 20% 20\% below average.\newlineWhich number type is more appropriate for the domain of H H ?\newlineChoose 11 answer:\newline(A) Integers\newline(B) Real numbers
Get tutor helpright-arrow

Posted 7 months ago

Question
The altitude at which we boil an egg affects how long it takes for the egg to achieve perfect hardness.\newlineIt takes 198198 seconds to boil a perfect egg at the lowest place possible, the edge of the Dead Sea, which has an altitude of −418-418 meters.\newlineThe highest place possible is the summit of Mount Everest which has an altitude of 88488848 meters. It takes 209209 seconds to boil a perfect egg there.\newlineT(a)T(a) models the time (in seconds) it takes to boil a perfect egg at an altitude of aa meters.\newlineWhich number type is more appropriate for the domain of TT?\newlineChoose 11 answer:\newline(A) Integers\newline(B) Real numbers
Get tutor helpright-arrow

Posted 7 months ago

Question
f(x)=x−1 f(x)=\sqrt{x-1} \newlineThe function f f is defined. What is the value of f(10) f(10) ?
Get tutor helpright-arrow

Posted 10 months ago

Question
h(x)=4x h(x)=\sqrt{4 x} \newlineThe function h h is defined. What is the value of h(9) h(9) ?\newlineChoose 11 answer:\newline(A) 33\newline(B) 66\newline(C) 1212\newline(D) 3636
Get tutor helpright-arrow

Posted 8 months ago

View all (64)

Related topics

Algebra - Order of OperationsAlgebra - Distributive Property`X` and `Y` AxesGeometry - Scalene TriangleCommon MultipleGeometry - Quadrant