5.2 Linear Equations in One Variable with Applications
3 min read•june 18, 2024
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
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- maintains equality when adding the same value to both sides of an equation ( is equivalent to )
- maintains equality when subtracting the same value from both sides of an equation ( is equivalent to )
- maintains equality when multiplying both sides of an equation by the same non-zero value ( is equivalent to )
- maintains equality when dividing both sides of an equation by the same non-zero value ( is equivalent to )
- Simplify each side of the equation by combining ( becomes )
- Isolate the term on one side of the equation using the ( becomes )
- Solve for the variable by performing the on both sides of the equation ( becomes )
Modeling applications with linear equations
- Identify the unknown quantity and assign a variable to represent it (let represent the number of apples purchased)
- Translate the given information into using the variable (cost of apples: , cost of oranges: )
- Construct a by setting the expressions equal to each other based on the problem context (total cost: )
- Solve the resulting linear equation using the properties of equality and algebraic techniques (, )
- 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
- (consistent and independent)
- After simplifying the equation, the variable term has a non-zero and is on one side of the equation, while the other side is a constant ( has one solution, )
- (inconsistent)
- After simplifying the equation, the variable terms cancel out, leaving a false statement ( simplifies to , which is false, so there is no solution)
- (consistent and dependent)
- After simplifying the equation, the variable terms cancel out, leaving a true statement ( simplifies to , which is true for any value of , so there are infinitely many solutions)
Linear Functions and Graphs
- A linear is a relationship between two variables where the rate of change () is constant
- The represents a linear function in the form y = mx + b, where m is the slope and b is the y-
- 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)
Key Terms to Review (25)
Addition Property of Equality: The addition property of equality states that if you add the same number to both sides of an equation, the two sides remain equal. This property is crucial for solving linear equations, as it allows us to isolate variables by performing equivalent operations on both sides of the equation, maintaining the integrity of the equation throughout the solving process.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression or equation. It indicates how many times to scale the variable, influencing the overall value of the expression. Understanding coefficients is crucial in various mathematical contexts, as they help to establish relationships between variables and define equations in both algebraic and linear formats.
Consistent Equation: A consistent equation is an equation that has at least one solution, meaning it can be satisfied by at least one value of the variable involved. This concept is important as it relates to the broader understanding of solutions in linear equations, where consistent equations can either have exactly one solution or infinitely many solutions depending on their form. Recognizing consistent equations helps in solving real-world problems modeled by linear relationships.
Dependent Equation: A dependent equation is a type of equation that has infinitely many solutions, meaning that every solution to the equation satisfies the original equation. This occurs when two or more linear equations represent the same line in a graph, leading to an overlap of solutions. Recognizing dependent equations is crucial when solving systems of linear equations, as it indicates that the equations are not independent and can be used interchangeably.
Division Property of Equality: The division property of equality states that if you divide both sides of an equation by the same non-zero number, the two sides remain equal. This fundamental principle allows us to isolate variables and solve linear equations, making it essential for working with equations in one variable and applying them in various real-world scenarios.
Equation of a line: An equation of a line is a mathematical expression that describes a straight line in a coordinate system, typically represented in slope-intercept form as $$y = mx + b$$, where $$m$$ is the slope and $$b$$ is the y-intercept. This equation illustrates how the variables change in relation to one another, allowing us to model real-world situations and solve problems involving linear relationships.
Formula: A formula is a mathematical expression that relates variables and constants in an equation to solve a problem or find an unknown value. In algebra, formulas are used to solve linear equations and model real-world situations.
Function: A function is a relationship or rule that assigns each input exactly one output. It can be represented using equations, tables, or graphs, and is fundamental in mathematics for modeling real-world situations and understanding relationships between quantities. Functions can be linear, where the relationship between variables creates a straight line when graphed, or non-linear, which may produce curves or more complex shapes.
Inconsistent equation: An inconsistent equation is a mathematical statement that has no solution because the equations represented by it cannot be satisfied simultaneously. This occurs when two or more linear equations contradict each other, leading to a situation where there is no value that can satisfy all the equations at once. Understanding inconsistent equations helps in identifying relationships between variables in real-world applications, highlighting scenarios where no common solution exists.
Independent Equation: An independent equation is a linear equation that has a unique solution and is not influenced by any other equations in a system. This means that the equation represents a line on a graph that intersects the x-axis and y-axis at specific points, determining a single unique point of intersection. Understanding independent equations is crucial for solving systems of equations, as they help establish conditions for the existence and uniqueness of solutions in various applications.
Infinitely many solutions: Infinitely many solutions refer to a situation in mathematical equations where there are countless valid solutions that satisfy the equation. This typically occurs in linear equations when the equations represent the same line or plane, leading to multiple points of intersection. Understanding this concept is crucial for solving and interpreting both single-variable and systems of linear equations.
Intercept: An intercept is a point where a line crosses an axis on a graph, typically represented in coordinate geometry as the x-intercept and y-intercept. The x-intercept is where the line intersects the x-axis (where y = 0), while the y-intercept is where the line intersects the y-axis (where x = 0). Understanding intercepts is crucial for analyzing linear equations and their graphical representations, as they provide key information about the behavior of the equation.
Inverse operation: An inverse operation is a mathematical process that reverses the effect of another operation, effectively allowing one to undo the previous operation. This concept is essential in solving linear equations, as it helps isolate the variable and find its value. Understanding inverse operations is crucial for performing arithmetic operations such as addition and subtraction, or multiplication and division, as they directly relate to manipulating equations in a way that reveals solutions.
Isolating Variables: Isolating variables is the process of manipulating an equation to get one variable alone on one side, typically to solve for that variable. This technique is crucial for simplifying linear equations, allowing for clearer solutions and better understanding of relationships between variables. It involves using algebraic operations such as addition, subtraction, multiplication, and division to achieve this goal.
Like Terms: Like terms are terms in an algebraic expression that have the same variable raised to the same power. They can be combined through addition or subtraction, making them crucial for simplifying expressions and solving equations. Identifying like terms is essential in algebra, as it helps in organizing expressions and performing calculations more efficiently.
Linear equation: A linear equation is a mathematical statement that describes a straight line when graphed on a coordinate plane, typically expressed in the form $$y = mx + b$$, where $$m$$ represents the slope and $$b$$ represents the y-intercept. This equation shows a direct relationship between two variables, indicating how one variable changes with respect to another. Linear equations can be manipulated and solved for unknowns, making them essential in various applications, from real-world problems to graphical representations.
Mathematical Expressions: Mathematical expressions are combinations of numbers, variables, and operations that represent a value or a relationship. These expressions serve as the foundational building blocks for solving equations, particularly linear equations in one variable, by providing a means to quantify and manipulate relationships between different quantities.
Modeling: Modeling is the process of representing real-world situations using mathematical expressions or equations, allowing us to analyze and make predictions about those situations. By simplifying complex scenarios into manageable forms, modeling helps us understand relationships between variables and solve problems effectively. This approach is essential for applying linear equations in one variable to various practical applications, such as economics, physics, or everyday life situations.
Multiplication property of equality: The multiplication property of equality states that if you multiply both sides of an equation by the same nonzero number, the two sides remain equal. This property is fundamental when solving linear equations in one variable, as it allows for the manipulation of equations while preserving their balance.
No solution: No solution refers to a situation in which an equation or system of equations has no set of values that satisfies it. This can occur when the equations represent parallel lines in a graph, indicating that they never intersect and thus have no common solutions. Understanding this concept is vital for recognizing when certain problems cannot be solved within the given constraints, whether it's a single equation or a system involving multiple equations.
One solution: One solution refers to a scenario in which a linear equation in one variable has exactly one unique answer. This means that when the equation is solved, it produces a specific value for the variable, indicating that there is a single point of intersection between the line represented by the equation and the x-axis. Understanding this concept is essential for analyzing equations and their applications, as it helps in determining the relationship between variables and identifying feasible solutions in real-world situations.
Properties of Equality: Properties of equality are fundamental rules that govern how to manipulate equations and maintain equality between both sides of an equation. They include the addition, subtraction, multiplication, and division properties, which allow for systematic solving of linear equations in one variable. Understanding these properties is essential for simplifying and rearranging equations to find unknown variables.
Slope: Slope is a measure of the steepness or incline of a line, typically represented as the ratio of the vertical change to the horizontal change between two points on that line. It plays a crucial role in understanding relationships in equations and inequalities, helping to determine whether they increase or decrease, and is essential for graphing functions and analyzing systems of equations.
Subtraction property of equality: The subtraction property of equality states that if two expressions are equal, subtracting the same number from both sides will maintain that equality. This property is fundamental in solving linear equations because it allows for the manipulation of equations to isolate variables, making it easier to find their values.
Variable: A variable is a symbol, typically a letter, that represents an unknown quantity in mathematical expressions and equations. Variables allow us to create general rules and relationships in mathematics, enabling the representation of multiple values without specifying them directly. They play a critical role in forming algebraic expressions and solving linear equations, as they can change or vary based on the context or specific problem at hand.