2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
Last Updated on June 24, 2024
Solving equations is a fundamental skill in algebra. It involves using properties of equality to isolate variables and find their values. This process is crucial for understanding more complex mathematical concepts and solving real-world problems.
The addition and subtraction properties of equality are key tools for solving linear equations. These properties allow us to manipulate equations while maintaining balance, helping us isolate variables and find solutions efficiently.
Solving Equations Using Subtraction and Addition Properties of Equality
Verification of linear equation solutions
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Understand the concept of a solution to an equation
Value that makes the equation true when substituted for the variable (x=3 in 2x+1=7)
Substitute the given value for the variable in the original equation
Simplify the left side of the equation by performing operations (addition, subtraction, multiplication, division)
Simplify the right side of the equation by performing operations
Check if the left side equals the right side
Equal sides confirm the given value is a solution (6+1=7)
Unequal sides indicate the given value is not a solution (6+1=8)
Properties for equation solving
Addition Property of Equality
Adding the same value to both sides of an equation maintains equality (a=b implies a+c=b+c)
Example: If x+3=7, then x+3−3=7−3 and x=4
Subtraction Property of Equality
Subtracting the same value from both sides of an equation maintains equality (a=b implies a−c=b−c)
Example: If x−5=2, then x−5+5=2+5 and x=7
Isolate the variable term on one side of the equation
Add or subtract the same value on both sides to eliminate the constant term on the variable side (x+3=7 becomes x=7−3)
Simplify both sides of the equation (x=4)
The value on the side without the variable is the solution
Equations with simplification steps
Simplify each side of the equation by combining like terms (2x+3x−4=5 becomes 5x−4=5)
Use the Distributive Property to remove parentheses, if necessary
a(b+c)=ab+ac (2(x+3)=10 becomes 2x+6=10)
Isolate the variable term on one side of the equation
Add or subtract the same value on both sides to eliminate the constant term on the variable side (5x−4=5 becomes 5x=9)
Simplify both sides of the equation (x=59)
The value on the side without the variable is the solution
Word problems to equations
Identify the unknown quantity and assign a variable to it (let x represent the unknown number)
Translate the word problem into an equation using the given information
Use key phrases to determine the operations and relationships between quantities
"sum," "more than," or "increased by" indicate addition (x+5=12)
"difference," "less than," or "decreased by" indicate subtraction (x−3=7)
"product," "times," or "multiplied by" indicate multiplication (2x=18)
"quotient," "divided by," or "ratio" indicate division (4x=6)
Write an equation that represents the problem ("5 more than an unknown number is 12" becomes x+5=12)
Solve the equation using the appropriate properties of equality (x=7)
Real-world applications of linear equations
Read the problem carefully and identify the given information and the question to be answered
Assign a variable to the unknown quantity (let x represent the number of hours worked)
Write an equation that represents the problem using the given information ("total pay is 15perhourplusa50 bonus" becomes 15x+50=290)
Solve the equation using the subtraction and addition properties of equality (x=16)
Interpret the solution in the context of the original problem (16 hours were worked)
Check if the solution makes sense in the given context
If not, review the problem-solving steps for errors (negative hours worked is not possible)
Fundamental Concepts in Equation Solving
Algebraic expressions: Combinations of variables, numbers, and operations (e.g., 2x + 3)
Equation solving: The process of finding the value(s) of a variable that make an equation true
Mathematical reasoning: Logical thinking used to analyze and solve mathematical problems
Number sense: Understanding of numbers, their relationships, and operations
Algebraic manipulation: Rearranging and simplifying algebraic expressions to solve equations
Key Terms to Review (17)
Variable: A variable is a symbol, usually a letter, that represents an unknown or changeable quantity in an algebraic expression or equation. It is a fundamental concept in algebra that allows for the representation and manipulation of unknown or varying values.
Constant: A constant is a fixed value or quantity that does not change within a given context or problem. It is a fundamental component in algebraic expressions and equations, providing a stable reference point for mathematical operations and calculations.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the number or quantity that is applied to the variable, indicating how many times the variable is to be used in the expression.
Algebraic Expressions: Algebraic expressions are mathematical representations that combine variables, constants, and operations to express relationships and quantities. They are a fundamental component of algebra, used to model and solve a variety of problems involving unknown values.
Subtraction Property of Equality: The subtraction property of equality states that if you subtract the same quantity from both sides of an equation, the equality will still hold true. This property allows you to isolate variables and solve for unknown values in an equation.
One-Step Equations: A one-step equation is a linear equation that can be solved in a single step, typically involving only one operation such as addition, subtraction, multiplication, or division. These equations are the simplest form of linear equations and are often the starting point for students learning to solve equations.
Isolate the Variable: Isolating the variable is the process of manipulating an equation to solve for a specific unknown variable by performing inverse operations to move all other variables and constants to one side of the equation. This technique is essential in solving linear equations and is a fundamental skill in elementary algebra.
Equal Sign: The equal sign (=) is a mathematical symbol used to indicate that two expressions or values are equivalent or have the same value. It is a fundamental concept in algebra that establishes a relationship of equality between the left and right sides of an equation.
Solution: A solution is a homogeneous mixture composed of two or more substances. In a solution, a solute is dissolved in a solvent, resulting in a single phase with a uniform composition and properties.
Linear Equations: A linear equation is a mathematical equation in which the variables are raised to the first power and the variables are connected by addition, subtraction, multiplication, or division. Linear equations represent straight-line relationships between variables and are fundamental in solving various algebraic problems.
Balancing Equations: Balancing equations is the process of adjusting the number of atoms of each element on the reactant and product sides of a chemical equation to ensure that the total number of atoms of each element is the same on both sides. This ensures that the law of conservation of mass is upheld, meaning that matter is neither created nor destroyed during a chemical reaction.
Equation Solving: Equation solving is the process of finding the value of an unknown variable in an equation by using mathematical operations and properties to isolate the variable. It is a fundamental skill in algebra that allows for the resolution of linear, quadratic, and other types of equations.
Inverse Operations: Inverse operations are mathematical operations that undo or reverse the effects of other operations. They are essential for solving equations and working with algebraic expressions by allowing you to isolate variables and find unknown values.
Mathematical Reasoning: Mathematical reasoning refers to the process of using logical and analytical thinking to solve mathematical problems, make mathematical connections, and justify mathematical conclusions. It involves the ability to understand, interpret, and apply mathematical concepts, principles, and procedures to various situations.
Number Sense: Number sense refers to a deep understanding of numbers, their relationships, and the ability to perform mental calculations and estimations flexibly and fluently. It is a critical foundation for success in mathematics, enabling individuals to develop a conceptual grasp of numerical concepts beyond mere memorization of facts and procedures.
Algebraic Manipulation: Algebraic manipulation refers to the process of performing various operations and transformations on algebraic expressions to simplify, solve, or rearrange them. It involves the strategic application of mathematical rules and properties to manipulate variables, coefficients, and exponents in order to achieve a desired form or result.
Addition Property of Equality: The addition property of equality states that if two expressions are equal, adding the same number to both sides of the equation will result in two new equal expressions. This fundamental principle allows for the manipulation of equations to isolate variables and solve for unknown values.