Master Algebra 1 Word Problems with our Comprehensive Practice

What is the missing sequence in ${\displaystyle 1\frac{3}{8},1\frac{3}{4},2\frac{1}{8}{,}_{\_},2\frac{7}{8}}$?

What is the ordered pair for y=4x+6 when x=−3?

Find the explicit formula for the following sequence 1, 10, 100, 1000, 10000,...

How do you determine if 15,-5,-25,-45 is an arithmetic or geometric sequence?

For which real values of t does the following system of linear equations:
$$\{\begin{array}{c}t{x}_1+x_2+x_3=1\\ x_1+t{x}_2+x_3=1\\ x_1+x_2+t{x}_3=1\end{array}$$
Have:
a) a unique solution?
b) infinitely many solutions?
c) no solutions?

How do you write the first five terms of the sequence defined recursively ${\displaystyle a_1=14,a_{k+1}=(-2)a_k}$, then how do you write the nth term of the sequence as a function of n?

Differential equation: the law of natural growth and the law of natural decay
I understand that $\frac{dy}{dx}=k\ast <!--\; \ast \; -->y$ and when k>0 this is the law of natural growth and when k<0 this is the law of natural decay, but my textbook gives an example of radioactive decay as follows which confuses me:
Radioactive substances decay by spontaneously emitting radiation. If is the mass remaining from an initial mass of the substance after time t, then the relative decay rate
$\frac{-<!--\; -\; -->1}{m}\frac{dm}{dt}$
has been found experimentally to be constant. (Since $\frac{dm}{dt}$ is negative, the relative decay rate is positive.) It follows that:
$\frac{dm}{dt}=km$
where k is a negative constant. In other words, radioactive substances decay at a rate proportional to the remaining mass. This means that we can use to show that the mass decays exponentially:
$m(t)=m(0)e^{kt}$ (This equation I understand and accept, the above two confuses me)
Now eventually when the example becomes numerical, then of course k becomes a negative number since it is natural decay. However, the above general notation explanation confuses me because it keeps changing the sign of k=relative decay rate(from negative(1) to positive(2), but eventually when numerically worked it turns out to be a negative constant since it's natural decay). I know that k must be less than zero for decay but I'm just trying to fully grasp the notation signs that the textbook uses in the explanation above.

y=(20x+3)r+x and y is 50. What is x and r values?

Find the slope of the line perpendicular to y=−3x+9

What number is 5 less than half of 32?

I want to find sum of 14 and the product of 8 and a number

The attendance at two baseball games on successive nights was 77,000. The attendance on​ Thursday's game was 7000 more than​ two-thirds of the attendance at Friday​ night's game. How many people attended the baseball game each​ night?

I'm trying to graph a simple response function: 1/(1-0.5s^-1)
Now, I know that the function can also be written as: s/(s-0.5)
So I tried plotting the step and impulse responses in Matlab:
sys = tf([1 0],[1 -0.5])
figure(1);
step(sys);
figure(2);
impulse(sys);
However, both graphs look the same (can't post images of my graphs, I need more rep to do it).
Both graphs have exponential growth, but shouldn't the impulse response look like exponential decay?

A colleague came across this terminology question.
What are the definitions of exponential growth and exponential decay? In particular:
1) Is $f(x)=-<!--\; -\; -->e^x$ exponential growth, decay, or neither?
2) Is $g(x)=-<!--\; -\; -->e^{-<!--\; -\; -->x}$ exponential growth, decay, or neither?
Consider $f(x)=A{e}^{kx}.$. I can't find any sources that specify A>0. My answer is that:
$f$ exhibits
1. exponential growth for A>0,k>0, and
2. exponential decay for A>0,k<0
whereas $|f|$ exhibits
3. exponential growth for A<0,k>0, and
4. exponential decay for A<0,k<0.
In case (3) we shouldn't call $f$ an exponential growth function without noting that it is "negative growth". Also it wouldn't be called it an exponential decay function without specifying the "direction of decay", so it is neither.
In case (4) it's neither as well. One should specify that it is the magnitude of $f$ which decays exponentially although $f$ is increasing in value. Although $f$ is increasing in value, is it growing? It seems odd to say it is exponentially growing.
It just doesn't sit right with me to refer to a function as growing if it is decreasing in value. Certainly, it's magnitude may be growing.
Next consider a function with exponential asymptotic behavior (e.g. logistic) so that as $x\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->},$ $f(x)\approx <!--\; \approx \; -->A{e}^{-<!--\; -\; -->kx}+C$ for some k>0. I feel the best way to describe this would be "exponential decay towards C" with a qualification as being from as being from above or below depending on the sign of A.
If someone is to just use the terminology "exponential growth (decay)", it implies $f(x)=A{e}^{kx}$ with positive A and k>0 (k<0) unless there is a specific context or further clarification as to what the actual nature of the function is.

Write the first six terms of the sequence ${\displaystyle a_n={(n-1)}^2}$

Find the 20th term of the arithmetic sequence 2, –4, –10, …

What is the solution for 4x-y=11?

How to solve a Non-algebraic equation?
Whilst working with exponential boom and decay, I encountered a hassle wherein I need to solve an equation related to logarithm. I could not separate or could not make it explicit.
$y=\frac{\ln <!--\; \; -->((1+r)(1-<!--\; -\; -->\mathrm{\delta <!--\; \delta \; -->}))}{\ln <!--\; \; -->((1+\frac{r}{y})(1-<!--\; -\; -->\mathrm{\delta <!--\; \delta \; -->}))}$. where r and δ are positive, both are smaller than 1.
I need the solution of this in the form of; $y=f(r,\mathrm{\delta <!--\; \delta \; -->})$. I tried numerical solution also but could not find a best way. Any hint or help is appreciated.