Linear Equations and First-Degree Equations with One Unknown

Linear Equations and Inequalities

Introduction

This section introduces the concept of linear equations, focusing on first-degree equations with one unknown. These equations are foundational in College Algebra and are essential for solving a wide range of mathematical problems.

First-Degree Equations with One Unknown

Definition and Structure

• Equation: An equation is a mathematical statement that asserts the equality of two expressions.

• First-degree equation with one unknown: An equation in which the variable appears to the first power and is the only variable present. Example: $x + 5 = 3x + 4$

• This type of equation is also called a linear equation with one unknown.

Key Properties

• The variable (e.g., x) has an exponent of 1.

• There is only one variable in the equation.

Example: Testing a Solution

• Given the equation $3a + 4 = a + 12$, test if $a = -2$ is a solution:

Substitute $a = -2$:

$3(-2) + 4 = -2 + 12$ $-6 + 4 = 10$ $-2 eq 10$

• Conclusion: $a = -2$ is not a solution.

Properties of Equivalent Equations

Definition

Equivalent equations are equations that have the same solution set. Certain operations can be performed on both sides of an equation without changing its solutions.

Properties

• If $a = b$, then $a + c = b + c$ and $a - c = b - c$.

• Adding or subtracting the same number from both sides yields an equivalent equation.

• If $a = b$ and $c eq 0$, then $ac = bc$ and $\frac{a}{c} = \frac{b}{c}$.

• Multiplying or dividing both sides by the same nonzero number yields an equivalent equation.

Example: Solving by Addition

• Solve $a - 6.2 = -3.5$:

Add $6.2$ to both sides:

$a - 6.2 + 6.2 = -3.5 + 6.2$ $a = 2.7$

• Check: $2.7 - 6.2 = -3.5$

Example: Solving by Multiplication

• Solve $\frac{1}{7}w = 3$:

Multiply both sides by $7$:

$7 \cdot \frac{1}{7}w = 3 \cdot 7$ $w = 21$

Procedure for Solving First-Degree Equations

Step-by-Step Method

1. Remove grouping symbols (parentheses) in the proper order.

2. If fractions exist, multiply both sides by the least common denominator (LCD).

3. Collect like terms, if possible.

4. Add or subtract variable terms on both sides to obtain all variable terms on one side.

5. Add or subtract numerical values on both sides to obtain all constants on the other side.

6. Divide both sides by the coefficient of the variable.

7. Simplify the solution, if possible.

8. Check your solution by substituting it back into the original equation.

Example: Multi-Step Solution

• Solve $x + 5 = 3x + 4$:

Subtract $3x$ from both sides: $x + 5 - 3x = 3x + 4 - 3x$ $-2x + 5 = 4$

Subtract $5$from both sides:$-2x = -1$

Divide both sides by $-2$: $x = \frac{1}{2}$

• Check: $\frac{1}{2} + 5 = 3 \cdot \frac{1}{2} + 4$ $5.5 = 1.5 + 4 = 5.5$

Special Cases in Linear Equations

No Solution

• An equation that is always false for any value of the variable has no solution.

• Example: $3c - 17 = 8c - 5(c - 2)$ simplifies to $-17 = 10$, which is never true.

Identity (Infinite Solutions)

• An equation that is always true for any value of the variable is called an identity and has infinitely many solutions.

• Example: $2k + 10 = 7k - 10$ after simplification may result in $0 = 0$.

Summary Table: Properties of Equivalent Equations

Operation	Result
Add/Subtract same number	Equivalent equation
Multiply/Divide by same nonzero number	Equivalent equation
Change variable side	Equivalent equation

Conclusion

Mastering the solution of first-degree equations with one unknown is essential for success in algebra. Understanding the properties of equivalent equations and following systematic procedures ensures accurate and efficient problem-solving.