3.5 Solve Equations Using Integers; The Division Property of Equality
3 min read•june 24, 2024
Solving equations with is all about finding the right numbers that make an equation true. It's like a puzzle where you use math properties to figure out what value fits perfectly in place of the .
The is a handy tool for solving equations. It's like sharing a pizza equally among friends - you divide both sides of the equation by the same number to keep things balanced and find the solution.
Solving Equations with Integers
Integer solutions for equations
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Substitute the given integer for the variable in the equation to check if it is a solution
Simplify both sides of the equation by performing the necessary operations (addition, subtraction, multiplication, division)
Compare the simplified left and right sides of the equation
Equal sides confirm the integer is a solution to the equation
Unequal sides indicate the integer is not a solution to the equation
Follow the (PEMDAS) when simplifying expressions
Properties for equation solving
adds the same number to both sides of an equation, maintaining equality
a=b implies a+c=b+c
subtracts the same number from both sides of an equation, maintaining equality
a=b implies a−c=b−c
Isolate the variable on one side by applying the addition or subtraction property of equality
Add or subtract the same value from both sides to eliminate the term on the variable side
Combine on each side of the equation to simplify
Use the to simplify expressions with parentheses
The Division Property of Equality
Division property in practice
Division Property of Equality divides both sides of an equation by the same non-zero number, maintaining equality
Sharing a pizza equally among friends demonstrates the division property
Mathematically, a=b and c=0 imply ca=cb
Scaling a recipe applies the division property to determine new ingredient amounts
1 cake2 cups of flour=3 cakesx cups of flour finds the flour needed for 3 cakes
The (reciprocal) is used when dividing by fractions
Solving equations with division
Isolate the variable on one side by dividing both sides by the of the variable
Divide both sides of the equation by the same non-zero number to eliminate the coefficient
Simplify the equation by performing the division on both sides
Check the solution by substituting the value back into the original equation to verify equality
Word problems to equations
Identify the unknown quantity and represent it with a variable
Translate the word problem into an equation using the given information and the variable
Ensure consistent units on both sides of the equation
Solve the equation using the appropriate properties of equality (addition, subtraction, division)
Interpret the solution in the context of the original word problem
Check if the solution makes sense in the given context (positive lengths, whole numbers for counts)
Types of Equations and Their Properties
involve variables and mathematical operations
are a type of algebraic equation where the variable has an exponent of 1
The distributive property is used to simplify expressions with parentheses in equations
Key Terms to Review (17)
#ERROR!: #ERROR! is a common error message that appears when a formula or function in a spreadsheet or other software application encounters an issue that prevents it from producing a valid result. This term is particularly relevant in the context of whole number operations, solving equations, and working with fractions, as these mathematical concepts are foundational to understanding and troubleshooting #ERROR! messages.
÷: The division symbol, also known as the obelus, represents the mathematical operation of division. It indicates that the number or expression to the left of the symbol is to be divided by the number or expression to the right of the symbol. This key term is crucial in understanding various mathematical concepts, including finding multiples and factors, solving equations using integers, and multiplying and dividing fractions and mixed numbers.
Addition Property of Equality: The addition property of equality states that if two expressions are equal, adding the same number to both expressions will result in two new expressions that are also equal. This property allows for the manipulation of equations by adding the same value to both sides to isolate a variable or solve for an unknown.
Algebraic Equations: Algebraic equations are mathematical statements that express the relationship between variables and constants using algebraic operations. They are fundamental to solving problems in pre-algebra and algebra, as they allow for the manipulation and simplification of expressions to find unknown values.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the scale or magnitude of the variable, indicating how much of that variable is present in the expression.
Constant: A constant is a quantity in an algebraic expression or equation that has a fixed, unchanging value. It is a value that does not vary or change throughout the given context or problem.
Distributive Property: The distributive property is a fundamental algebraic rule that states the product of a number and a sum is equal to the sum of the individual products. It allows for the simplification of expressions by distributing a factor across multiple terms within a parenthesis or other grouping symbol.
Division Property of Equality: The division property of equality states that if two expressions are equal, and you divide both sides by the same non-zero number, the resulting expressions will still be equal. This property allows you to isolate variables in an equation by dividing both sides by the coefficient of the variable.
Integers: Integers are a set of positive and negative whole numbers, including zero. They are the foundation for many mathematical operations and concepts, and are essential in understanding and working with various topics in pre-algebra.
Inverse Operations: Inverse operations are mathematical operations that undo or reverse the effects of another operation. They are used to solve equations, simplify expressions, and perform calculations in various mathematical contexts.
Isolating the Variable: Isolating the variable is the process of manipulating an equation to solve for a specific unknown or variable. This technique is essential in solving linear equations and is a fundamental skill in algebra and pre-algebra courses.
Like Terms: Like terms are algebraic expressions that have the same variable(s) raised to the same power. They can be combined by adding or subtracting their coefficients, allowing for the simplification of algebraic expressions.
Linear Equations: A linear equation is a mathematical equation in which the variables are raised to the power of one. These equations represent a straight line when graphed, and they are characterized by a constant rate of change between the variables.
Multiplicative Inverse: The multiplicative inverse of a number is the reciprocal of that number, which when multiplied with the original number, results in the multiplicative identity of 1. The multiplicative inverse is a crucial concept in various mathematical operations and equations involving whole numbers, integers, fractions, and decimals.
Order of Operations: The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed to evaluate an expression. This term is crucial in the context of evaluating, simplifying, and translating expressions, as well as solving equations using various properties of equality.
Subtraction Property of Equality: The subtraction property of equality states that if the same number is subtracted from both sides of an equation, the equality is still maintained. This property allows for the simplification and solving of linear equations by isolating the variable on one side of the equation.
Variable: A variable is a symbol, typically 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 generalization of mathematical relationships and the solution of problems involving unknown values.