Solving One-Variable Equations

Solving one-variable equations is a fundamental skill in algebra, enabling you to find the value of an unknown variable that satisfies the equation. A one-variable equation typically involves only one unknown (usually denoted by x) and is expressed as a mathematical statement with an equal sign. The goal is to isolate the variable on one side of the equation to determine its value.

Types of One-Variable Equations

One-variable equations come in many forms, but they can usually be classified into three main categories:

1. Linear Equations: These equations involve the variable raised to the first power (e.g., 2x+5=152x + 5 = 15).
2. Quadratic Equations involve the variable raised to the second power (e.g., x2+3x−4=0x^2 + 3x - 4 = 0).
3. Rational Equations: These equations involve fractions with the variable in the numerator or denominator (e.g., 1x+2=5\frac{1}{x} + 2 = 5).

However, the most common type of equation you will encounter in introductory algebra is a linear equation. Linear equations have the general form:

ax+b=cax + b = c Where:

• x is the unknown variable,
• a and b are constants (numbers),
• c is another constant.

The objective is to solve for x by isolating it on one side of the equation.

Steps for Solving One-Variable Equations

Solving one-variable equations involves applying basic algebraic operations to isolate the variable. Here’s a step-by-step guide for solving simple linear equations:

Step 1: Simplify Both Sides

Before solving for the variable, simplify the equation by combining like terms. This might involve adding or subtracting terms, or distributing values across parentheses.

For example, consider the equation:

3(x+4)=183(x + 4) = 18First, distribute the 3 on the left side:

3x+12=183x + 12 = 18Step 2: Eliminate Constants from One Side

Next, you need to eliminate the constants from the side where the variable is not located. To do this, use inverse operations. In the above example, subtract 12 from both sides:

3x=63x = 6Step 3: Isolate the Variable

Now, you can isolate the variable by performing the inverse operation of the coefficient of x. In this case, divide both sides of the equation by 3:

x=63x = \frac{6}{3}x=2x = 2Thus, the solution to the equation is x = 2.

Step 4: Verify the Solution

Always check your solution by substituting the value of x back into the original equation to ensure that both sides are equal. For the example above, substitute x = 2 into the original equation:

3(2+4)=18⇒3(6)=18⇒18=183(2 + 4) = 18 \quad \Right arrow \quad 3(6) = 18 \quad \Right arrow \quad 18 = 18Since both sides are equal, the solution x = 2 is correct.

Solving More Complex One-Variable Equations

One-variable equations can become more complex, but the same principles apply. You may encounter equations with fractions, decimals, or multiple terms on either side. In these cases, follow these strategies:

1. Precise Fractions: Multiply both sides of the equation by the least common denominator (LCD) to eliminate fractions.
2. Precise Decimals: Multiply both sides by 10, 100, or another power of 10 to eliminate decimals.
3. Distribute: If there are parentheses, use the distributive property to expand them.

For example, to solve 12x+3=7\frac{1}{2}x + 3 = 7, multiply both sides by 2 to eliminate the fraction:

x+6=14x + 6 = 14Now subtract six from both sides:

x=8x = 8

Solving one-variable equations is a foundational skill in algebra. You can isolate the variable and find its value by applying basic operations like addition, subtraction, multiplication, and division. Whether dealing with simple linear equations or more complex problems involving fractions and decimals, the key to solving one-variable equations is to systematically isolate the variable through inverse operations. Always remember to verify your solution to ensure its correctness.