代数概念中的直觉

Algebra is the art and process of solving for an unknown variable (In Arabic, al-jebr means "the reunion of broken parts"). Here are some basic techniques for solving.

Strategy 1: Isolation.

Isolation is where you isolate the variable (the value you are solving for) and the constant (a known value).

Take a look at the example:

Say you have two variables and two equations.What would you do?

Strategy 2: Substitution.

Substitution is what it sounds like. You substitute a given value to reduce the system of equations to one variable, one equation. Here is an example.

Last strategy!

Strategy 3: Elimination

Again, we deal with 2 variables, 2 equations. We first manipulate the equations (multiply or divide both sides by a constant) so if we add or subtract the equations, we eliminate a variable.

Notice we could have multiplied the second equation by 4 and found x first to plug it into the first equation. We would've gotten the same answer. We could have also substituted; there are many ways to solve for variables. Which one do you use? There are some methods that are more efficient at times, but less efficient at other times. But at the end, it's all your choice! $\ddot \smile$

Note

I feel that this topic has been avoided for a while, because Algebra I is a very wide subject, but here it is in a nutshell.

Algebra

Algebra is a branch of mathematics that deals with unknown values represented by variables. These variables can include letters like $a, b, c, x, y, z$ and possibly other special characters (Greek) like $\alpha$ (alpha), $\theta$ (theta), or $\mu$ (mu). We usually want to find these values, using an equation, which states something about a variable that can help with finding it. Solving the equation, you can determine the identity of a variable. Really simple equations are like $x=9$, which just gives the value. There's no complicated solving. more complicated equations are like $3x+2=6x-7$. Let's walk through this equation, noting the techniques we may have learned in pre-algebra: $$3x+2=6x-7$$ $$3x+2-(3x)=6x-7-(3x)$$ $$2=3x-7$$ $$2+7=3x-7+7$$ $$9=3x$$ $$3=x$$

Lets take solving equations to the next level. We now have quadratic equations of the form $ax^2+bx+c$=0. Let's start with an example. $$x^2+3x+5=3$$ $$x^2+3x+2=0$$, which after factoring goes to $$(x+1)(x+2)=0$$. In order for $$(x+1)(x+2)$$ to equal $0$, either $(x+1)$ or $(x+2)$ has to equal 0. This, using our linear equation techniques, $x=-1$ or $-2$.

Graphs of Equations

We can graph these equations on a coordinate plane (the extended $xy$ plane, not the small ones that we used in elementary school). The graphs of linear equations are lines (this is in their name), and the graphs of parabolas are parabolas. We graph the equations $y=x$ (a line) and $y=x^2$ (a parabola). Try to tell which is which.

Functions are special graphs. A function essentially spits out one and only one value of $y$ for each value of $x$. The Vertical Line test determines if an equation is a function. If any vertical line intersects the graph at two or more points (meaning two or more values of $y$ for that value of $x$) then that equation is not a function. Instead of using $y$ for functions you can use $f(x)$, $g(x)$, $\gamma(x)$, etc.

That basically is a general explanation of Algebra I. Feel free to add another explanation to this topic.