back to Algebra

One variable first degree equation

Expression with an Equal Sign

The expression use the equal sign "=" to express the equal relationship between left side of the equal sign and right side of the equal sign.

Example

Equation Property 1

Add the same number or the same polynomial on both side of the equal sign, the final result still satisfy the left side of equal sign is equal to the right side of the equal sign.

Example

Look the example shown below, when both side of the equation add the same value, the left side of the equal sign still equals to the right side of the equal sign. Keep the variable term in the left side of the equation.

Equation Property 2

Subtract the same number or the same polynomial on both side of the equal sign, the final result still satisfy the left side of equal sign is equal to the right side of the equal sign.

Example

Equation Property 3

Multiply the same number or the same polynomial on both side of the equal sign, the final result still satisfy the left side of equal sign is equal to the right side of the equal sign.

Example

Equation Property 4

Divide the same number or the same polynomial on both side of the equal sign, the denominator can not be zero, the final result still satisfy the left side of equal sign is equal to the right side of the equal sign.

Example

Known Number

That the value of the variable is given in the question is called known number.

Examples

Joe is 12 years now, J = 12 is a given number.

Unknown Number

That the value of the variable is not known and you need to find it in the question is called unknown number.

Examples

Joe is 12 years now, Steven is 5 years more than Joe, how old is Steven now? S = ? , S is a unknown number.

The Example of what is the unknown number.

Equation

The equation is constructed from unknown variable and known number.

Example

5x + 8 = 9 is an equation, since it contain veriable x which is unknown. 7 + 8 = 15 is not an equation, since it does not have any unknown variable.

What is an equation? what is a known number and what is an unknown number?

Solution of the Equation

The value of the variable that let the left side of the equal sign is equal to the right side of the equal sign is called solution of the equation.

Example

Move all variable terms to the left side of the equation, and move all the constant terms to the right side of the equation, then you get the value of x, by substituting the value of x to the original equation to verify whether your solution is correct.

One Variable First Degree Equation

The equation that has only one variable with degree of one is called one variable one degree equation. Its standard syntax is ax + b = 0 (a is not zero).

Example

Look the example shown below, left side of the equation has a constant of 8, when move the constant of 8 to the right side of the equation, the 8 is changed to - 8 (its sign was changed to opposite sign).

The steps how to solve an one variable first degree equation.

Moving Terms

Moving a term from a side of the equal sign to its opposite side of the equal sign.

Example

In equation 3x + 5 = 2x + 10, move term 2x from right side of the equal sign to left side of the equal sign, move the term 5 from left side of the equal sign to right side of the equal sign. Note: after moing, the sign of the term is changed to its opposite sign.

Rules of Moving Term

Change the sign of the term to its opposite sign when moving it to other side of the equal sign.

Example

In equation 3x + 5 = -2x +10, after moving term -2x from right side to left side of the equal sign, the -2x change to +2x, after moving +5 from left side to right side of the equal sign, the +5 is changed to -5.

Steps to Solve the One Variable First Degree Equation

Step 1: Remove denominator by using common denominator.

Step 2: Remove parenthesis by using distributive property a(b + c) = ab + ac.

Step 3: Moving terms with unknown variable to the left side of the equal sign and terms with known number to the right side of the equal sign.

Step 4: Combining similar terms together to make this syntax, ax = b.

Step 5: Make coefficient of the variable to 1 to get the final value of the variable x = b/a (a is not zero).

Example

$Example of how to solve a fraction equation.$

The least common denominator(LCD) is 6, so both side of the equation times 6, then we get an one degree and one variable equation.

Example of Solving Problem 1

After school start, John spent half of his saving to buy books, then he received $20 from his mother, he found that he had a total of $50 left, how much did John has before school start?

Analysis: (1). Let x is the original dollar he had before school start.; (2). He spent (1/2)x to buy books.; (3). He received $20 from his mother.; (4). He has a total of $50 left.; (5). Get the equation for this problen, detail solution in right side, John has $60 before school start.

Solution: $Example of how to solve a fraction equation.$

Example of Solving Problem 2

A stock increase 35% in yr. 2000, then it decrease 75% to dollar b in yr. 2001, what is the price of this stock before yr. 2000?

Analysis: (1). Let x is the price of this stock before yr. 2000.; (2). In yr. 2000, the price of this stock is (1 + 35%) x; (3). In yr. 2001, the price of this stock is (1 - 75%) (1 + 35%) x = b; (4). Solve the equation; (5). the price of this stock before yr. 2000 is 2.96 the price in yr. 2001.

Solution