Linear Equations in One Variable and the Addition Property of Equality

Linear Equations in One Variable

Definition and Identification

Linear equations in one variable are fundamental to algebra. They are equations that can be written in the standard form, where the variable appears only to the first power and is not multiplied by itself or any other variable.

• Definition: A linear equation in one variable x can be written as $ax + b = c$, where a, b, and c are real numbers, and a \neq 0.

• First Degree: Linear equations are called first degree equations because the highest power of the variable is one.

• Examples:

  • $3x + 5 = 11$ is linear because it fits the form $ax + b = c$.

  • $|x| + 2 = 7$ is not linear because of the absolute value around x.

  • $x - 4 = 10$ is linear; it can be written as $1x + (-4) = 10$.

The Addition Property of Equality

Solving Linear Equations

The addition property of equality is a key principle used to solve equations. It states that adding (or subtracting) the same value to both sides of an equation does not change the solution.

• Property Statement: If $a = b$, then $a + c = b + c$ for any real number or algebraic expression c.

• Application: This property allows us to isolate the variable by adding or subtracting terms from both sides.

• Example: Solve $x - 9 = 12$.

  • Add 9 to both sides: $x - 9 + 9 = 12 + 9$

  • Simplifies to $x = 21$

  • The solution set is {21}

• Subtraction: Subtracting a number from both sides is equivalent to adding its opposite.

  • Example: Solve $x + 6 = 15$

  • Subtract 6: $x + 6 - 6 = 15 - 6$

  • $x = 9$

Adding and Subtracting Variable Terms

Sometimes, variable terms appear on both sides of the equation. The goal is to collect all variable terms on one side and constants on the other.

• Example: Solve $2x + 3 = x + 5$

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

  • $x + 3 = 5$

  • Subtract 3: $x = 2$

  • Check: Substitute $x = 2$ into the original equation: $2(2) + 3 = 2 + 5 \rightarrow 4 + 3 = 7 \rightarrow 2 + 5 = 7$; both sides equal 7.

Solving and Checking Solutions

After solving an equation, it is important to check the solution by substituting it back into the original equation.

• Example: Solve and check $x - 4 = 13$

  • Add 4: $x = 17$

  • Check: $17 - 4 = 13$

• Example: Solve and check $x + 5 = 12$

  • Subtract 5: $x = 7$

  • Check: $7 + 5 = 12$

Applied Problems Using Formulas

Modeling Real-World Relationships

Linear equations can be used to model real-world situations, such as the relationship between a child's vocabulary and age.

• Example: The number of words in a child's vocabulary, $V$, and the child's age, $A$, in months (for ages 15 to 50 months) can be modeled by:

  • $V = 40A + 100$

  • To find the vocabulary at 50 months: $V = 40 \times 50 + 100$

  • $V = 2000 + 100 = 2100$

  • At 50 months, a child will have a vocabulary of 2100 words.

Summary Table: Linear Equation Identification

The following table summarizes how to identify whether an equation is linear:

Equation	Linear?	Reason
$3x + 5 = 11$	Yes	Fits $ax + b = c$ form
$|x| + 2 = 7$	No	Absolute value makes it nonlinear
$x - 4 = 10$	Yes	Can be written as $1x + (-4) = 10$
$x^2 + 3 = 7$	No	Variable is squared (degree 2)
$2x + 3 = x + 5$	Yes	Both sides linear in $x$

Additional info: Some examples and explanations were inferred for completeness and clarity, especially where original content was fragmented or missing.