Linear equations are the building blocks of algebra. They help us solve real-world problems by turning words into math. We use properties like addition and multiplication to simplify equations and find solutions.

Applications of linear equations are everywhere. We can model situations, from figuring out how many apples we bought to calculating complex business decisions. The key is translating the problem into an equation and solving it step by step.

Solving Linear Equations and Applications

Properties and techniques for linear equations

• Addition property of equality maintains equality when adding the same value to both sides of an equation ($x + 3 = 5$ is equivalent to $x + 3 + 2 = 5 + 2$)
• Subtraction property of equality maintains equality when subtracting the same value from both sides of an equation ($x - 4 = 7$ is equivalent to $x - 4 - 1 = 7 - 1$)
• Multiplication property of equality maintains equality when multiplying both sides of an equation by the same non-zero value ($2x = 10$ is equivalent to $2x \cdot 3 = 10 \cdot 3$)
• Division property of equality maintains equality when dividing both sides of an equation by the same non-zero value ($3x = 12$ is equivalent to $\frac{3x}{3} = \frac{12}{3}$)
• Simplify each side of the equation by combining like terms ($2x + 3x - 4 = 6$ becomes $5x - 4 = 6$)
• Isolate the variable term on one side of the equation using the properties of equality ($5x - 4 = 6$ becomes $5x = 10$)
• Solve for the variable by performing the inverse operation on both sides of the equation ($5x = 10$ becomes $x = 2$)

Modeling applications with linear equations

• Identify the unknown quantity and assign a variable to represent it (let $x$ represent the number of apples purchased)
• Translate the given information into mathematical expressions using the variable (cost of apples: $2x$, cost of oranges: $3(5) = 15$)
• Construct a linear equation by setting the expressions equal to each other based on the problem context (total cost: $2x + 15 = 35$)
• Solve the resulting linear equation using the properties of equality and algebraic techniques ($2x = 20$, $x = 10$)
• Interpret the solution in the context of the original problem, ensuring it makes sense and answers the question asked (10 apples were purchased)

Solutions of linear equations

• One solution (consistent and independent)
  • After simplifying the equation, the variable term has a non-zero coefficient and is on one side of the equation, while the other side is a constant ($2x + 3 = 7$ has one solution, $x = 2$)
• No solution (inconsistent)
  • After simplifying the equation, the variable terms cancel out, leaving a false statement ($3x - 6 = 3x + 2$ simplifies to $0 = 8$, which is false, so there is no solution)
• Infinitely many solutions (consistent and dependent)
  • After simplifying the equation, the variable terms cancel out, leaving a true statement ($4x + 2 = 4x + 2$ simplifies to $0 = 0$, which is true for any value of $x$, so there are infinitely many solutions)

Linear Functions and Graphs

• A linear function is a relationship between two variables where the rate of change (slope) is constant
• The equation of a line represents a linear function in the form y = mx + b, where m is the slope and b is the y-intercept
• The slope measures the steepness of the line and represents the rate of change between variables
• The y-intercept is the point where the line crosses the y-axis (when x = 0)