I. Intro to Equation
A linear equation is an equation that represents a straight line when graphed on a coordinate plane. The highest power of the variable in a linear equation is always 1.
The standard form of a linear equation in one variable is:
ax + b = 0
II. Solving Equations
Sometimes when we have an algebraic expression in an equation, we would like to find a number or set of numbers that make the equation a true statement. That is, we want to find a number (or numbers) for the variable(s) which makes both sides of the equation equal. This number is called a solution to the equation and the process of finding the number is called solving (a set of numbers is called the solution set and the elements of it are called solutions).Equality
Equality is a fundamental idea of algebra. It is indicated by the equal sign and means that two mathematical expressions have the same value. One way to think about equality (and equations) is as a balanced scale. As in the picture below, we can change the number of ovals on each side of the scale and it still remains balanced.
Properties of Equality
Properties of equality describe the relation between two equal quantities and that if an operation is applied on one side of the equation, then it must be applied on the other side to keep the equation balanced.
The addition property of equality and the subtraction property of equality state that if we add or subtract the same number on both sides of an equation, then the resulting sides of the equation will remain equal.
When the equation involves addition or subtraction, use the inverse operation to “undo” the operation in order to isolate the variable. For addition and subtraction, your goal is to change any value being added or subtracted to 0.
Example:
x - 6 = 8
x - 6 + 6 = 8 + 6x = 14
Using the Addition Property of Equality, 6 was added to both sides of the equation to isolate the variable.
Using the Addition Property of Equality, 6 was added to both sides of the equation to isolate the variable.
Example:
3,500 + 4,800 + x = 10,000
8,300 + x = 10,000
8,300 - 8,300 + x = 10,000 - 8,300
x = 1,700
The multiplication property of equality states that if both the sides of an equation are multiplied by the same number, the expressions on the both sides of the equation remain equal to each other. The division property of equality tells us that if we divide both sides of an equation by the same number, the equation remains the same.
When the equation involves multiplication or division, you can “undo” these operations by using the inverse operation to isolate the variable. When the operation is multiplication or division, your goal is to change the coefficient to 1, the multiplicative identity.
Example:
3x = 24
3x/3 = 24/3
x = 8
Using the division property of equality, both sides of the equation are divided (inverse of multiplication) by 3 to isolate the variable.
Example:
Using the multiplication property of equality, both sides of the equation are multiplied by 2 to isolate the variable.
Example:
-7/2 = k/10
(10)-7/2 = k/10(10)
-35 = k
C. Modeling With One-Step Equations (Word Problems)
D. Dependent & Independent Variables
A dependent variable is a variable whose value changes based on the value of another variable. It is the effect value. An independent variable is a variable that is being manipulated. It is the cause value. Example:
You are doing chores to earn your allowance. For each chore you do, you earn $3. We can write this as an equation where C = each individual chore and M = total money earned.
M = 3C
The dependent variable is M, the amount of money you earn because it is dependent on how many chores you do. The independent variable is the amount of chores you do because this is the variable you have control over.
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