# Tutor profile: Ravi M.

## Questions

### Subject: Pre-Calculus

Solve the following equation for $$x$$. $$\log_{6}{48x^3}-\log_{6}{2x}=3$$

Using the rule of logarithms, $$\log_{c}{a}-\log_{c}{b}=log_{c}{\frac{a}{b}}$$ Hence $$\log_{6}{48x^3}-\log_{6}{2x}=log_{6}{\frac{48x^3}{2x}}$$ (Note that $$48x^3>0$$ and $$2x>0$$ in order for the logarithms to be defined. Therefore, $$x>0$$) Therefore $$log_{6}{\frac{48x^3}{2x}}=3$$ $$log_{6}{24x^2}=3$$ $$6^{log_{6}{24x^2}}=6^3$$ $$24x^2=216$$ $$x^2=9$$ $$x=3$$ or $$x=-3$$, but remember that $$x>0$$, therefore the only solution is $$x=3$$.

### Subject: Calculus

Evaluate the following integral: $$\int 7x\cos(\frac{x}{3})dx$$

Let's use integration by parts to evaluate the integral. Remember the formula: $$\int udv = uv-\int vdu$$ Let's choose $$u=7x$$ and $$dv=\cos(\frac{x}{3})dx$$ Therefore $$u=7x$$ $$du=7dx$$ $$v=3\sin(\frac{x}{3})$$ $$dv=\cos(\frac{x}{3})dx$$ Then the original integral becomes the expression $$(7x)(3\sin(\frac{x}{3}))-\int(3\sin(\frac{x}{3}))(7dx)$$ $$21x\sin(\frac{x}{3})-21\int\sin(\frac{x}{3})dx$$ $$21x\sin(\frac{x}{3})-21(-3\cos(\frac{x}{3}))+c$$ $$21x\sin(\frac{x}{3})+63\cos(\frac{x}{3})+c$$

### Subject: Algebra

Solve the following system of equations for $$x$$ and $$y$$ : $$3x+5y=22$$ $$4x+2y=6$$

Let's solve for y in the second equation We have: $$4x+2y=6$$ $$4x+2y-4x=6-4x$$ $$2y=6-4x$$ $$\frac{2y}{2}=\frac{6-4x}{2}$$ $$y=\frac{6}{2} - \frac{4x}{2}$$ $$y=3-2x$$ Now using substitution, we can replace $$y$$ in the first equation. We have: $$3x+5y=22$$ and $$y=3-2x$$ By substitution, we have $$3x+5(3-2x)=22$$ $$3x+5(3) + 5(-2x)=22$$ $$3x+15-10x=22$$ $$-7x+15=22$$ $$-7x+15-15=22-15$$ $$-7x=7$$ $$\frac{-7x}{-7}=\frac{7}{-7}$$ $$x=-1$$ Now let's solve for $$y$$ by plugging $$x=-1$$ into $$y=3-2x$$ $$y=3-2x$$ $$y=3-2(-1)$$ $$y=3-(-2)$$ $$y=3+2$$ $$y=5$$ Remember to double check the solution by plugging them into the original equations: $$3x+5y=22$$ $$3(-1)+5(5)=22$$ $$-3+25=22$$ $$22=22$$ and $$4x+2y=6$$ $$4(-1)+2(5)=6$$ $$-4+10=6$$ $$6=6$$ Therefore, the solution checks and we have $$x=-1$$ and $$y=5$$

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