+
+
+
+ Welcome to a basic introduction of algebra. In this tutorial, we will
+ review some of the more common algebraic concepts.
+
+
+
+
+
+ To start with, let's define some of the more common terms used in
+ algebra:
+
+
+ -
+ Variable: A variable is an unknown value, usually represented
+ by a letter.
+
+ -
+ Expression: Essentially a mathematical object. For the
+ purpose of this tutorial, an expression is one part of an equation.
+
+ -
+ Equation: An equation is a mathematical argument in which two
+ expressions result in the same value.
+
+
+
+
+
+
+ Sometimes it is easier to understand the definitions when you have a
+ physical example to look at. Here is an example of the above terms.
+ x + 5 = 12
+ In this above example, we have:
+
+
+ - Variable: The variable in the example is "x".
+ -
+ Expression: There are two expressions in this example. They
+ are "x+5" and "12".
+
+ -
+ Equation: The entire example, "x+5=12", is an equation.
+
+
+
+
+
+
+ The primary use for algebra is to determine an unknown value, the
+ "variable", with the information provided. Continuing to use our
+ example from above, we can find the value of the variable "x".
+ x + 5 = 12
+ In an equation, both sides result in the same value. So you can
+ manipulate the two expressions however you need, as long as you
+ perform the same operation (or change) to each side. You do this
+ because the goal when solving an equation is to
+ get the variable into its own expression, or by itself on one side
+ of the = sign.
For this example, we want to remove the "+5" so the "x" is
+ alone. To do this, we can subtract 5, because subtraction is
+ the opposite operation to addition. But remember, we have to perform
+ the same operation to both sides of the equation. Now our equation
+ looks like this.
+ x + 5 - 5 = 12 - 5
+ The equation looks like a mess right now, because we haven't completed
+ the operations. We can simplify this equation to make it easier
+ to read by performing the operations "5-5" and "12-5". The result
+ is:
+ x = 7
+ We now have our solution to this equation!
+
+
+
+
+
+ Let us look at a slightly more challenging equation.
+ 3x + 4 = 13
+ Again we can start with subtraction. In this case, we want to subtract
+ 4 from each side of the equation. We will also go ahead and simplify
+ with each step. So now we have:
+ 3x = 9
+ "3x" translates to "3*x", where the "*" symbol indicates
+ multiplication. We use the "*" to avoid confusion, as the "x" is now a
+ variable instead of a multiplication symbol. The opposite operation
+ for multiplication is division, so we need to
+ divide each expression by 3.
+ x = 3
+ And now we have our solution!
+
+
+
+
+
+ Now we are getting in to more complex operations. Here is another
+ equation for us to look at:
+ x^2 - 8 = 8
+ Our very first step will be to add 8 to each side. This is
+ different from our previous examples, where we had to subtract. But
+ remember, our goal is to get the variable alone by performing opposite
+ operations.
+ x^2 = 16
+ But what does the "^2" mean? The "^" symbol is used to denote
+ exponents in situations where superscript is not available. When
+ superscript is available, you would see it as x2.
+ For the sake of this project, however, we will use the "^" symbol.
+ An exponent tells you how many times the base (in our case, "x") is
+ multiplied by itself. So, "x^2" would be the same as "x*x". Now the
+ opposite function of multiplication is division, but we would have to
+ divide both sides by "x". We do not want to do this, as that
+ would put an "x" on the other side of the equation. So instead, we
+ need to use the root operation! For an exponent of "2", we call this
+ the "square root" and denote it with "√". Our equation is now:
+
+ x = √9
+ Performing a root operation by hand can be a tedious process, so we
+ recommend using a calculator when necessary. However, we are lucky in
+ that "9" is a
+ perfect square, so we do not need to calculate anything. Instead, we find our
+ answer to be:
+ x = 3
+
+
+
+
+
+ As you explore your algebra studies further, you may start to run
+ across equations with more than one variable. The first such equations
+ will likely look like:
+ y = 3x
+ An equation like this does not have one single solution.
+ Rather, there are a series of values for which the equation is true.
+ For example, if "x=3" and "y=9", the equation is true. These equations
+ are usually used to plot a graph.
+ Getting more complicated, though, you may be given a pair of
+ equations. This is called a "system of equations", and CAN be solved.
+ Let's look at how we do this! Consider the following system of
+ equations:
+ y = 3x | y - 6 = x
+ A system of equations IS solvable, but it is a multi-step process. To
+ get started, we need to chose a variable we are solving for. Let's
+ solve for "x" first. From the second equation, we know that "x" equals
+ "y - 6", but we cannot simplify that further because we do not have a
+ value for "y". Except, thanks to the system of equations, we DO have a
+ value for "y". We know that "y" equals "3x". So, looking at our second
+ equation, we can replace "y" with "3x" because they have the same
+ value. We then get:
+ 3x - 6 = x
+ Now we can solve for "x"! We start by adding 6 to each side.
+ 3x = x + 6
+ We still need to get "x" by itself, so we subtract "x" from both sides
+ and get:
+ 2x = 6
+ If this confuses you, remember that "3x" is the same as "x+x+x".
+ Subtract an "x" from that and you get "x+x", or "2x". Now we divide
+ both sides by 2 and have our value for x!
+ x = 3
+ However, our work is not done yet. We still need to find the value for
+ "y". Let's go back to our first equation:
+ y = 3x
+ We have a value for "x" now, so let's see what happens if we put that
+ value in.
+ y = 3*3
+ We perform the multiplication and discover that "y=9"! Our solution to
+ this system of equations then is:
+ x = 3 and y = 9
+
+
+
+
+ Coming Soon!
+ Keep an eye out for new additions!
+
+
+
+ Check out the following links for more information!
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