Solving Linear Equations: Properties of Equality and Applications

Equations, Inequalities, and Problem Solving

Linear Equations in One Variable

Linear equations in one variable are equations that can be written in the form ax + b = c, where a, b, and c are real numbers and x is the variable. The solution to a linear equation is the value of the variable that makes the equation true.

• Equation: A statement that two algebraic expressions are equal.

• Expression: A mathematical phrase that does not include an equality sign.

• Solution Set: The set of all values that satisfy the equation, written in set brackets { }.

Example: Determine whether the given value is a solution to the equation = . Substitute x = -3:

$2(-3) + 6 = -6 + 6 = 0$

Since the statement is true, x = -3 is a solution.

Identifying Linear Equations

Not all equations are linear. A linear equation in one variable has the variable raised only to the first power and does not involve products of variables or higher powers.

• Linear: $ax + b = c$

• Nonlinear: $x^2 = 25$ or $2(x + 5) - 3x = x^2 + 1$

Example: Which of the following is a linear equation in one variable?

Addition and Subtraction Properties of Equality

To solve linear equations, we use the properties of equality. These properties state that the same operation can be performed on both sides of an equation without changing the solution set.

• Addition Property: If $a = b$, then $a + c = b + c$

• Subtraction Property: If $a = b$, then $a - c = b - c$

These properties are used to isolate the variable on one side of the equation.

Solving Linear Equations: Step-by-Step

To solve a linear equation, follow these steps:

1. Simplify both sides of the equation (distribute, combine like terms).

2. Use addition or subtraction to collect all variable terms on one side and constants on the other.

3. Use multiplication or division to isolate the variable.

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

Example: Solve $2x + 6 = 0$

$2x + 6 = 0$ $2x = -6$ $x = -3$

Visualizing Equations

Equations can be represented visually using a balance scale, where each side of the equation is balanced.

Practice Problems

• Solve $x - 6 = 0$

• Solve $0 = x + 2$

• Solve $y - 1.2 = 5.8$

• Solve $m - 9 = -6$

• Solve $x - (-5) = 12$

These problems reinforce the use of addition and subtraction properties to isolate the variable and solve for its value.

Summary Table: Linear vs. Nonlinear Equations

Type	Example	Degree
Linear	$2x + 6 = 0$	1
Nonlinear	$x^2 = 25$	2

Additional info: The images included above are directly relevant to the explanation of linear equations, properties of equality, and the process of solving equations. They visually reinforce the algebraic steps and concepts discussed in the text.