# 1.2 – Solving Linear Equations

## Objectives

• Review steps for solving linear equations, including collecting like terms, using the order of operations, and performing inverse operations.
• Identify equations that have no solutions or infinitely many solutions.
• Ask the questions necessary to turn real-life problems into mathematical sentences.
• Identify constraints that might affect the possible solutions in a real-life problem.

## Key Terms

• Distributive Property – the rule that if a, b, and c are numbers or expressions, then a (b + c) = a b + a c.
• Infinitely – without end or limit; going on forever; impossible to count.
• Inverse Operations – an inverse operation is an operation that reverses (undoes) the action of another operation.
• Like Terms – terms in an algebraic expression that have the same variables raised to the same powers.
• Linear Equation – any equation whose graph is a straight line. A linear equation can be written in the form y = mx + b.
• Standard Form of a Linear Equation – a linear equation written in the form Ax + By + C = 0.

## Notes

Distributive Property
• Question: When do I need to use the distributive property?
• Answer: When a variable is part of an expression inside parentheses that is multiplied by a number or variable, like 3(x + 4).
• Question: Why do I need to use the distributive property?
• Answer: So I can collect like terms and solve an equation.
Standard Form of a Linear Equation
• a, b, and c are numbers & x is a variable
• ax + b = c
• ax – b = c
Solutions
• No Solution – you will end up with an expression that is never true, no matter what the value of x is.
• x = x + 7 (subtract x from both sides)
• 0 = 7 (not true)

• Infinitely Many Solutions – you will end up with an equation that is always true, no matter what the value of x is.
• 3(x + 2) = 3x + 6 (distribute 3)
• 3x + 6 = 3x + 6 (subtract 3x from both sides)
• 6 = 6 (always true)

• What is the solution?
• Step 1: Write Equation => 2x + 2 = 10
• Step 2: Isolate “x” using inverse operations, starting with constants => Subtract 2 on both sides (2x + 2 – 2 = 8 – 2)
• Step 3: Isolate “x” using inverse operations, remove coefficients => divide by 2 on both sides:
• $\frac{2x}{2}\ = \frac{4}{2}$