This lesson introduces an important algebraic concept known as the Equation. The idea is that an equation represents a scale such as the one shown on the right. Instead of keeping the scale balanced with weights, numbers or constants are used. These numbers are called constants because they constantly have the same value. For example the number 47 always represents 47 units or 47 multiplied by an unknown number. It never represents another value.
The equation may also be balanced by a device called a variable. A variable is an an unknown number represented by any letter in the alphabet (often x). The value of each variable must remain the same in each problem.
Several symbols are used to relate all of the variables and constants together. These symbols are listed and explained below.
+ Add or Positive
- Subtract or Negative
( ) Calculate what is inside of the parentheses first. (also called grouping symbols)
An expression in mathematics refers to a combination of mathematical symbols and operators and operators. For example,
5 + 3
is an expression since it combines the symbols 5 and 3 (in this case the symbols are real numbers) and the mathematical addition operator (+).
An expression can also consist of variables as in the example below
x - 12
where x is a variable because it represents a value which we do not know at the moment.
Expressions can also be written without operators as long as variables are involved, for example, the following are also expressions:
where t, x, y and α are variables.
Basics of the Equation
The diagram above shows a basic equation. This equation is similar to problems which you may have done in ordinary mathematics such as:
__ + 16 = 30
You could easily guess that __ equals 14 or do 30 - 16 to find that __ equals 14.
In this problem __ stood for an unknown number; in an equation we use variables, or any letter in the alphabet.
When written algebraically the problem would be:
x + 16 = 30
and the answer should be written:
x = 14
These equations can be solved relatively easy and without any formal method. But, as you use equations to solve more complex problems, you will want an easier way to solve them.
Pretend you have a scale like the one shown. On the right side there are 45 pennies and on the left side are 23 pennies and an unknown amount of pennies. The scale is balanced, therefore, we know that there must be an equal amount of weight on each side.
As long as the same operation (addition, subtraction, multiplication, etc.) is done to both sides of the scale, it will remain balanced. To find the unknown amount of pennies of the left side, remove 23 pennies from each side of the scale. This action keeps the scale balanced and isolates the unknown amount. Since the weight (amount of pennies) on both sides of the scale are still equal and the unknown amount is alone, we now know that the unknown amount of pennies on the left side is the same as the remaining amount (22 pennies) on the right side.
Because an equation represents a scale, it can also be manipulated like one. The diagram below shows a simple equation and the steps to solving it.
Initial Equation / Problem: x + 23 = 45
Subtract 23 from each side: x + 23 - 23 = 45 - 23
Result / Answer: x = 22
The diagram below shows a more complex equation. This equation has both a constant and a variable on each side. Again, to solve this you must keep both sides of the equation equal; perform the same operation on each side to get the variable "x" alone. The steps to solving the equation are shown below.
Initial Equation / Problem: x + 23 = 2x + 45
Subtract x from each side: x - x + 23 = 2x - x + 45
Result: 23 = x + 45
Subtract 45 from each side: 23 - 45 = x + 45 - 45
Result: -22 = x
Answer: x = -22
Take a look at the equation below. As you can see, after the variable is subtracted from the left and the constants are subtracted from the right, you are still left with 2x on one side.
Initial Equation / Problem: x + 23 = 3x + 45
Subtract x from each side: x - x + 23 = 3x - x + 45
Result: 23 = 2x + 45
Subtract 45 from each side: 23 - 45 = 2x + 45 - 45
Result: -22 = 2x
Switch the left and right sides of the equation: 2x = -22
This means that the unknown number multiplied by two, equals -22. To find the value of x, use the process "dividing by the coefficient".
Identifying and Using Coefficients
The coefficient of a variable is the number which the variable is being multiplied by. In this equation, 2 is the coefficient of x because 2x is present in the equation. Some additional examples of coefficients:
Term Coefficient of x
Note that in the last two examples, the following rules are applied
- If the variable has no visible coefficient, then it has an implied coefficient of 1.
- If the variable only has a negative sign, then it has an implied coefficient of -1.
Recall beginning to solve the equation "x + 23 = 3x + 45". Applying addition and subtraction gave
2x = -22
But our end goal is to determine what x is, not what 2x is! Imagine that three investors own an equal share in the company Example.Com. The total worth of Example.com is $300,000. To determine what the share of each investor is, simply divide the total investment by 3:
$300,000 / 3 = $100,000
Thus, each investor has a $100,000 stake in Example.com. We apply the same idea to finding the value of x. However, instead of dividing by the number of investors, we divide by the coefficient of the variable. Since we determined that the coefficient of x is 2, we divide each side of the equation by 2:
After dividing by 2: 1x = -11
Finally rewritten as: x = -11
Solving Equations https://www.mathsisfun.com/algebra/equations-solving.html
- Solving Simple Equations (lots of examples) https://www.cliffsnotes.com/study-guides/basic-math/basic-math-and-pre-algebra/variables-algebraic-expressions-and-simple-equations/solving-simple-equations
- Basic Algebra: An Introduction http://www.onlinemathlearning.com/basic-algebra.html
- Algebra 1 Worksheets: Equations https://www.kutasoftware.com/free.html
- Equations Worksheets (lots of good options) http://www.math-aids.com/Algebra/Algebra_1/Equations/
- One-Step, Two-Step, Multi-step Equations Worksheets https://www.mathworksheets4kids.com/equations/index.php
- Expressing Equations Worksheet http://www.commoncoresheets.com/Math/Algebra/Expressing%20Equations/English/1.pdf
- Solving Equations https://youtu.be/Oof-36FR_9o
- Linear Equations 1 https://www.khanacademy.org/math/algebra-home/alg-basic-eq-ineq/alg-old-school-equations/v/algebra-linear-equations-1