asinus

Give that the line y=4x-14 passes through the point (k,k+1), find the value of k.​

Hello Guest!

\(y=4x-14\\ k+1=4k-14\\ 3k=15\)

\(k=5\)

\(The\ line\ y=4x-14\ passes\ through\ the\ Point\ (5,6).\)

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Find the original fraction.

Hello Guest!

\(\dfrac{n}{d}=x\\ \dfrac{x-2}{d}=x\\ d=\dfrac{x-2}{x}\\ \dfrac{x-2}{\dfrac{x-2}{x}}=x\\ \dfrac{(x-2)+3}{(\dfrac{x-2}{x})+3}=\dfrac{19x}{18}\)

\(x=\dfrac{28}{29}\)

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Wieviel Geld hat jeder ausgeliehen?

Hallo Gast!

0,04a + 0,05b = 448

0,05b + 0,06c = 456

0,04a + 0,06c = 452

\(0,04a = 448-0,05b\\ 0,04a=452-0,06c\\ 448-0,05b=452-0,06c\ |\ Gleichsetzungsverfahren\\ 0,05b-0,06c=448-452\\ 0,05b-0,06c=-4\ |\ \cdot (-1)\\ \underline {0,05b + 0,06c = 456}\ |\ Additionsverfahren\)   

\(0,12c=460\)

\(c=3833,33\)€

\(0,05b + 0,06c = 456\\ 0,05b + 0,06\cdot 3833,33 = 456\\ 0,05b=226\)

\(b=4520\)€

\(0,04a + 0,06c = 452\\ 0,04a + 0,06\cdot 3833,33 = 452\\ 0,04a=222\)

\(a=5550\)€

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In  triangle ABC, CA=4sqrt(2), CB=4sqrt(7), and A=60 degrees. What is B in degrees?

Hello Guest!

sine law:

\(\frac{sinB}{\overline {CA}}=\frac{sinA}{\overline{CB}}\)

\(\frac{sinB}{4\sqrt{2} }=\frac{sin60°}{4\sqrt{7}}\\ \frac{sinB}{4\sqrt{2} }=\frac{\frac{1}{2}\sqrt{3} }{4\sqrt{7}}\\\)

\(sinB=\dfrac{\frac{1}{2}\sqrt{3}\cdot4\sqrt{2}}{4\sqrt{7}}=\dfrac{1}{2}\sqrt{\frac{6}{7}}\\ \angle B=asin\ (\dfrac{1}{2}\sqrt{\frac{6}{7}})\)

\(\angle B=27.575°\)

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What real value of t produces the smallest value of the quadratic t^2 - 9t - 36 + t^2 + 7t - 16?

Hello Guest!

\(f(t)=2t^2 - 2t - 52\\ \frac{df(t)}{dt}=4t-2=0\\ 4t=2\\ \color{blue}t=\frac{1}{2}\)

\(t=\frac{1}{2}\) produces the smallest value of the quadratic t^2 - 9t - 36 + t^2 + 7t - 16.

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The graph of y = f(x) is a parabola whose vertex is at (1,-2). The graph of y = f(x + 4) - 3 is also a parabola. Where is the vertex of the graph of y = f(x + 4) - 3?

Hello Guest!

The graph of  \(y=f(x)=( x-1)^2-2\) is a parabola whose vertex is at (1,-2).

The graph of y = f(x + 4) - 3 or  \(y=f(x)=(x-1{\color{blue}+4})^2-2-\color{blue}3=(x+3)^2-5\)  is also a parabola.

whose vertex  is at (-3,-5).

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As shown in the figure, two bisectors of ABC, BE and CF, intersect at P. If EPF = 111 °, what is \(\angle A \) in degrees?

Hello Guest!

\(\dfrac{180°-\alpha }{2}+\dfrac{180°-111°}{2}\color{BrickRed}=90°\ |\ mistake!\\ 180°-\alpha +180°-111°=2\cdot 90°\\ \alpha=180° +180°-111°-2\cdot 90° \)

\(\alpha \neq69°\)

\(\frac{\angle B}{2}+\frac{\angle C}{2}+111°=180°\)

\(\frac{\angle B}{2}+\frac{\angle C}{2}=180°-111°\\ (\angle B+\angle C)=2\cdot (180°-111°)\\ (\angle B+\angle C)+\angle A=180°\\ \angle A=180°-(\angle B+\angle C)\\ \angle A=180°-2\cdot (180°-111°)\)

\(\angle A=42° \)

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Find the value of  a so that the system has no solutions.

Hello Guest!

\(2x - 4 = ax + 7 - 3x\\ ax-3x-2x=-4-7\\ ax-5x=-11\\ x(a-5)=11\\ x=\dfrac{11}{a-5}\)

 The system has no solutions if a=5.

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6xy+10x2y=?

Hello Guest!

You can only factoring out.

\(6xy+10x^2y=2xy\cdot (3+5x) \)

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Robert is twice as old as Charlotte. Five years ago his age was three times Charlotte's age. Find their present ages.

Hello Guest!

\(r=2c\\ r-5=3(c-5)\\ 2c-5=3c-15\\ \color{blue}c=10\)

Charlotte is 10 years old.

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