Date modified: Wed 2023-08-23 . 02:04 PM
Date created: Sat 2023-10-07 . 09:02 PM
# Some algebra practice. Algebraic manipulations can be quite important. Although mathematically one is "not changing the equation", or rather, we are maintaining the relations, while performing algebraic manipulations, we can often re-write an expression in a more useful way, or solving for a certain quantity. You can do it! ## Problems. 1. Consider the equation:$$ A = \frac{B+1}{A+1} $$ - Solve for $B$ as an expression of $A$. - Solve for $A$ as an expression of $B$. Account for all solutions. 2. Consider the following equation: $$ T(S-U) + \frac{1}{S+U}=T^{2}U+SU^{2} $$Solve for $S$ as an expression of other variables. Give all solutions if multiple ones. 3. Consider the following equation: $$ B = A + \frac{1}{A + \frac{1}{A+B}} $$ Solve for $B$ as an expression of $A$. Give all solutions. 4. What if the expression is $$ A = A + \frac{1}{A + \frac{1}{A+B}} $$Can we solve for $B$ as an expression of $A$? Or rather, is there a $B$ for above equation to make sense? 5. Consider the following system of equations: $$ \begin{align*} 6A + 5B & =20 \\ A - 2B & = 30 \end{align*} $$ Solve for $A$ and $B$ exactly. Yes, answers can be fractions, they are numbers. 6. Consider the following system of equations: $$ \begin{align*} S & = S_{1} + S_{2} \\ T & = \frac{Q}{S-S_{1}} + k \frac{S_{1}}{S_{2}} \end{align*} $$Solve for $S_1$ and $S_2$ as expressions in $k, S, T, Q$ only. That is solve for $S_1$ without $S_2$ in it, and solve for $S_2$ without $S_1$ in it. 7. It is quite often useful to **rationalize** an expression, if you have a fraction involving radicals of the form $a + b \sqrt{c}$. Then we can often use its conjugate $a-b\sqrt{c}$ to simplify or re-express it, in the from of multiplying by $1 = \frac{a-b\sqrt{c}}{a-b\sqrt{c}}$. Also, the expression $\sqrt{a}+b$ has the conjugate to $\sqrt{a}-b$ as well. - Find the following limit by using the conjugate and algebra: $$ \lim_{x\to 0} \frac{x}{\sqrt{1+3x}-1} $$ - Find the following limit by somehow using the conjugates: $$ \lim_{x\to 2} \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1} $$Hint: You may need to use several conjugates... - As the limits above are all of the $\frac{0}{0}$ indeterminant forms, try verifying with L'Hospital rule as well. 8. Consider a trapezoid with height $H$, and the two parallel base sides $B_1$ and $B_2$. If we write $A$ as the area of this trapezoid, solve for $B_1$ as an expression of $A,H,B_2$. 9. Suppose a circular cylinder (with both top and bottom) has total surface area $100\text{ units}^{2}$, and we denote $R$ as the radius of the base circle of this cylinder, and $H$ as the height of this cylinder. Expression the height $H$ in terms of the radius $R$ of this cylinder. (You might want to figure out what the surface area expression is!) 10. A sequence of numbers is said to be an **arithmetic sequence** if two successive terms in this sequence has a **common difference**. For example $4, 15, 26, 37, 48$ is an arithmetic sequence with common difference $11$. In general, the common difference is $\text{next term} -\text{current term}$. - Suppose we have an infinite arithmetic sequence where parts of it reads $$ \ldots 64, 29, -6,\ldots $$and you are told that $64$ is the $5$-th term and $29$ is the $6$-th term of this arithmetic sequence, and so on. Find a formula for the $N$-th term of this arithmetic sequence. - Using your formula, what is the $1$-st term of this sequence? And what is the $100$-th term of this sequence? 11. A sequence of numbers is said to be a **geometric sequence** if two successive terms in this sequence has a **common ratio**. For example $2, 14, 98, 686$ is a geometric sequence with common ratio $7$. In general, the common ratio is $\frac{\text{next term}}{\text{current term}}$. - Suppose we have an infinite geometric sequence where parts of it reads $$ \ldots 5, 15, 45,\ldots $$and you are told that $5$ is the $21$-st term and $15$ is the $22$-nd term of this geometric sequence, and so on. Find a formula for the $N$-th term of this geometric sequence. - Using your formula, what is the $1$-st term of this sequence? And what is the $100$-th term of this sequence?