Solving Linear Equations Using Properties of Equality

Equations, Inequalities, and Problem Solving

Solving Linear Equations

Solving linear equations is a fundamental skill in intermediate algebra. The process involves applying algebraic properties to isolate the variable and verify the solution.

• Distributive Property: Used to remove parentheses by multiplying each term inside by the factor outside.

• Combining Like Terms: Simplifies the equation by adding or subtracting terms with the same variable or constant.

• Addition Property of Equality: Allows moving terms from one side of the equation to the other by adding or subtracting the same value from both sides.

• Division Property of Equality: Used to isolate the variable by dividing both sides by the coefficient of the variable.

Step-by-Step Example

Consider the equation:

$-5(3y - 5) - y = -5(y - 2)$

1. Apply the distributive property:

   • $-5(3y - 5) - y = -5(y - 2)$

   • $-15y + 25 - y = -5y + 10$

2. Combine like terms on the left side:

   • $-15y + 25 - y = -5y + 10$

   • $-16y + 25 = -5y + 10$

3. Use the addition property of equality:

   • Add $5y$ to both sides:

   • $-16y + 25 + 5y = -5y + 10 + 5y$

   • $-11y + 25 = 10$

   • Subtract $25$ from both sides:

   • $-11y = -15$

4. Divide both sides by $-11$:

   • $y = \frac{15}{11}$

5. Check the solution:

   • Substitute $y = \frac{15}{11}$ back into the original equation to verify that both sides are equal.

Key Terms

• Linear Equation: An equation involving only the first power of the variable.

• Distributive Property: $a(b + c) = ab + ac$

• Like Terms: Terms that have the same variable raised to the same power.

• Properties of Equality: Rules that allow manipulation of equations to maintain equality.

Example Application

Solving equations like this is essential for algebraic problem solving, including applications in science, engineering, and everyday life.