Master Algebra 1 Word Problems with our Comprehensive Practice

Consider the equation ${\displaystyle x+3=3(x+1)-2x}$. Is this equation a linear equation?
I would say to my students that an equation in 1 variable is linear when it can be simplified to the form ${\displaystyle ax=b}$ where ${\displaystyle a\ne 0}$. Also, I would mention that a linear equation in 1 variable has only 1 solution. Therefore, I would say that the equation ${\displaystyle x+3=3(x+1)-2x}$ is not linear because it can be reduced to ${\displaystyle 0x=0}$ which has infinite solutions.
However, a colleague whose research area is algebra says that a linear equation is one that only involves polynomials of degree 1. Therefore, the equation ${\displaystyle x+3=3(x+1)-2x}$ is linear. Then, according to the definition of my colleague, the equation ${\displaystyle x-x=0}$ is linear as well.
I was thinking that maybe a third possibility is that according to the definition of "linear equation" only equations of the form "expression=0" can be classified as linear or non-linear. In this case, the equation ${\displaystyle x+3=3(x+1)-2x}$ is not linear nor non-linear, but the equation ${\displaystyle x-x=0}$ would be linear.
What should be the definition for "linear equation" in 1 variable suitable for a math course?

In Linear Algebra Done Right, it defines the linear maps of a homogeneous system of linear equations with n variables and m equations as
${\displaystyle T(x_1,\dots ,x_n)=(\sum _{k=1}^n{A}_{1,k}{x}_k,\dots ,\sum _{k=1}^n{A}_{m,k}{x}_k)=0}$
My question is why the homogeneous system of linear equations can be expressed as the linear maps from ${\displaystyle {\mathbb{F}}^n\mapsto {\mathbb{F}}^m}$

Ms. Ragounath drove over a nail and the tire of her car started to leak air. If the car lost .4% of its air daily, when will her tire have three-quarters of its original air pressure which was 38 psi?

What is the solution to the system of equations below?
2x+3y=17
3x+6y=30

How many five-letter sequences are possible that use the letters c, p, c, p, z

Whats 5 times 5 and the quadratic formula to 24 3

write the standard form of a quadratic equation with roots of -6 and 4

How can I write a function that describes a relationship between two quantities, and what are some examples?

write an equation of the line that passes through (0, 5) and (2, -5)

Simplify each radical expression and perform the indicated operations. 6√32 − 2√8

x=4y 3x-2y=20

4(x + 2) This is my question. Please help me answer it!!!

The complex zeros of ${\displaystyle f\left(x\right)=x^4+1}$ For the function ${\displaystyle f\left(x\right)=x^4+1}$ :
(a) Factor f into the product of two irreducible quadratics. (Hint: Complete the square by adding and subtracting${\displaystyle 2x^2}$.)
(b) Find the zeros of f by finding the zeros of each irreducible quadratic.

How do you write a polynomial that represents the volume of a box that is a rectangular prism has the dimensions length ${\displaystyle x+6}$, width ${\displaystyle x-2}$, and height ${\displaystyle x-1}$?

1. Looking around in your own country, discuss two practical applications in which “piecewise functions” are used either explicitly or implicitly.
2. To the best of your ability, using mathematical notion or in words, create any one of the functions you have spoken of in part (1) above.
3. If you could only learn one approach to solving quadratic equations, discuss which of the approaches you would choose and why?
4. Do you wholly agree with the statement that “the domain and range of a quadratic equation will be ${\displaystyle [-\propto ,\propto ]}$ ”? Provide arguments to support your position

How do you write an expression to represent the total volume of two different sized boxes as a sum of cubes if one of the boxes has sides with a length of 1 foot and the others has sides with a length of x feet?

${\displaystyle e^{\left|\sin x\right|}+e^{-\left|\sin x\right|}+4a=0}$ will have exactly four different solution in ${\displaystyle [0,2\pi ]}$.
A) ${\displaystyle a\in [\frac{-e}{4},\frac{-1}{4}]}$
B) ${\displaystyle a\in [\frac{-1-e^2}{4e},\mathrm{\infty })}$
C) ${\displaystyle a\in \mathbb{R}}$
D) None of these.

Steepest descent (gradient method) for quadratic function
Let the quadratic function $F:{\mathbb{R}}^n⟶\mathbb{R}$ defined by:
${\displaystyle F\left(\mathbf{x}\right)=\frac{1}{2}⟨A\mathbf{x},\mathbf{x}⟩-⟨\mathbf{c},\mathbf{x}⟩,\mathrm{\forall }\mathbf{x}\in {\mathbb{R}}^n}$
where A is a real symmetric positive definite (square) matrix, ${\displaystyle c\in {\mathbb{R}}^n}$.

If k is even number then for what value of x and y,
${\displaystyle x^{2k}-{\left(xy\right)}^k+y^{2k}=0}$

Solve the equation ${\displaystyle \sin x+\cos x=k\sin x\cos x}$ for real x, where k is a real constant.