How to Solve Equations With Variables on Both Sides: Definition, Properties, and Solved Examples

Solving equations with variables on both sides is a fundamental skill in algebra that involves finding the value of the unknown variable that makes the equation true. This process often includes simplifying the equation, combining like terms, and applying algebraic properties such as the distributive property and the properties of equality. By learning the systematic approach to solving these equations with variables on both sides, you can tackle more complex mathematical problems with confidence. In this guide, we will explore the definition of these equations, review important properties and techniques, and work through solved examples to understand the concepts.

What is Equation?

An equation is a mathematical statement that asserts the equality of two expressions, separated by an equals sign (“=“). Each side of the equation consists of one or more terms involving numbers, variables, and operations like addition, subtraction, multiplication, and division.

Examples of equations include:

1. Simple Linear Equation:
2x+3 = 7
In this equation, 2x+3 and 7 are two expressions set equal to each other. The goal is to solve for x.
2. Equation with Multiple Variables:
3x−2y = 10
This equation involves two variables, x and y, and we look for solutions that satisfy the equality.
x2−4 = 0
This equation involves a squared term and requires solving for x such that the equation holds true.
4. Equation with Fractions:
(½)x+3 = 5
Here, the equation involves a fraction, and solving it requires handling the fractional term to isolate x.

What are Variables?

Variables are symbols used in mathematics to represent unknown values or quantities. They can take on different values depending on the context of the problem. Variables are often represented by letters such as x, y, or z, but any symbol can be used.

1. Placeholders for Values: Variables act as placeholders for values that are not yet known or are variable in a given situation.

For example, in the equation x+5 =, x is a variable that represents an unknown number.

1. Representation in Equations: In equations, variables are used to formulate relationships and solve for unknowns.

For example, in the equation, 2x−3 = 7, x is the variable we need to solve for.

1. Different Types of Variables:
• Independent Variables: Variables that are manipulated or changed in an experiment or function, often denoted as x.
• Dependent Variables: Variables that depend on the value of the independent variable, often denoted as y.
1. Algebraic Expressions: Variables can be part of algebraic expressions, such as 3x+2, where they are combined with constants and operators.
1. General Use: Variables can also be used in functions, sequences, and other mathematical contexts to represent a range of possible values.

Properties of Variables in Equations

In equations, variables have several important properties and characteristics that play an important role in solving mathematical problems. Here are some of the important properties of variables in equations:

1. Substitution Property: If two expressions are equal, then one can be substituted for the other in any equation.

For example, if x=5 and y=x, then y=5 can be substituted into any equation where y appears.

1. Isolating the Variable: To solve an equation, you often need to isolate the variable on one side of the equation. This involves using operations to move terms around and simplify the equation.

For example, in 2x+3 = 7, subtracting 3 from both sides gives 2x = 4, and then dividing both sides by 2 isolates x, resulting in x = 2.

1. Transposition: This property involves moving terms from one side of the equation to the other by applying inverse operations.

For example, if you have x+4 = 10, you can transpose 4 to the other side by subtracting 4 from both sides to get x = 6.

1. Combining Like Terms: Variables of the same kind (i.e., those with the same base and exponent) can be combined to simplify expressions.

For example, in 3x+2x = 10, the like terms 3x and 2x can be combined to give 5x = 10.

1. Distributive Property: When a variable is multiplied by a sum or difference, the distributive property can be used to expand the expression.

For example, a(b+c) = ab+ac. This property helps simplify equations and solve for the variable.

1. Equality of Expressions: If two expressions involving variables are equal, any operation performed on one side must be performed on the other side to maintain equality.

For example, if x+2 = y, then adding or subtracting the same value from both x+2 and y will keep the equation balanced.

1. Properties of Equality:

Equations With Variables on Both Sides Solved Examples

Here are five solved examples of equations with variables on both sides:

Example 1:

Equation: 3x+5 = 2x+8

Solution:

1. Subtract 2x from both sides to get the variable terms on one side:

3x+5−2x = 2x+8−2x

x+5=8

1. Subtract 5 from both sides to isolate xxx:

X+5−5 = 8−5

Example 2:

Equation: 4y−3 = 2y+9

Solution:

1. Subtract 2y from both sides to consolidate the y terms:

4y−2y−3 = 2y−2y+9

2y−3 = 9

1. Add 3 to both sides to isolate the y term:

2y−3+3 = 9+3

1. Divide both sides by 2 to solve for y:

2y/2 = 12/2

Example 3:

Equation: (¾)x−2 = (½)x+3

Solution:

1. Subtract (½)x from both sides to get the x terms together:

(¾)x−(½)x−2 = (½)x−(½)x+3

(¼)​x−2 = 3

1. Add 2 to both sides to isolate the x term:

(¼)x−2+2 = 3+2

1. Multiply both sides by 4 to solve for x:

4×(¼)x = 5×4

Example 4:

Equation: 7−2x = 3x+8

Solution:

1. Add 2x to both sides to consolidate the x terms:

7−2x+2x = 3x+2x+8

7 = 5x+8

1. Subtract 8 from both sides to isolate the xxx term:

7−8 = 5x+8−8

−1 = 5x

1. Divide both sides by 5 to solve for xxx:

−1/5 = x

X = −1/5​

Example 5:

Equation: 6−(⅔)x = (4/3)x+2

Solution:

1. Add (⅔)x to both sides to get the x terms together:

6−(⅔)x+(⅔)x = (4/3)x+(⅔)x+2

6 = (6/3)x+2

6 = 2x+2

1. Subtract 2 from both sides to isolate the x term:

6−2 = 2x+2−2

4=2x

1. Divide both sides by 2 to solve for x:

4/2 = 2x/2

x=2

FAQs

How do I solve an equation with variables on both sides?

To solve an equation with variables on both sides:
1.Combine like terms on each side.
2.Move variable terms to one side and constants to the other using inverse operations.
3.Isolate the variable by dividing both sides by the coefficient.

How to solve equations with variables on both sides using distributive property?

How to solve equations with variables on both sides using distributive property?

1. Distribute: Apply the distributive property to remove parentheses if present.2. Combine like terms: Simplify both sides of the equation by combining similar terms.3. Isolate variable: Move variable terms to one side and constants to the other using inverse operations.4. Solve for variable: Divide both sides by the coefficient of the variable to find its value.

How to solve an equation with two variables?

To solve an equation with two variables, you need two equations. Use methods like substitution (isolate one variable and replace it in the other equation) or elimination (add or subtract equations to cancel a variable) to find the values of both variables that satisfy both equations simultaneously.

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