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.
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The addition property of equality allows you to add the same number to both sides of an equation without changing the solution.
This property is commonly used to isolate a variable on one side of the equation by adding the opposite of the coefficient of the variable to both sides.
The addition property of equality is a fundamental tool in solving linear equations, as it enables you to perform the same operations on both sides of the equation to find the unknown value.
When solving equations using the addition property, it is important to ensure that the same value is added to both sides to maintain the equality.
The addition property of equality is applicable to equations involving integers, fractions, and decimals, making it a versatile tool in solving a wide range of algebraic equations.
Review Questions
Explain how the addition property of equality can be used to solve linear equations.
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 is commonly used in solving linear equations by isolating the variable on one side of the equation. For example, if you have the equation $2x + 5 = 11$, you can add the opposite of the coefficient of the variable ($-5$) to both sides to isolate the variable: $2x + 5 - 5 = 11 - 5$, which simplifies to $2x = 6$. By applying the addition property, you can perform the same operations on both sides of the equation to find the value of the unknown variable.
Describe how the addition property of equality is used in conjunction with the subtraction and division properties to solve more complex equations.
The addition property of equality is often used in combination with the subtraction and division properties of equality to solve more complex equations. For instance, if you have an equation with variables on both sides, such as $3x + 4 = 2x + 9$, you can first use the subtraction property to isolate the variable terms on one side: $3x + 4 - 2x = 2x + 9 - 2x$, which simplifies to $x + 4 = 9$. Then, you can apply the addition property to isolate the variable by adding the opposite of the constant term ($-4$) to both sides: $x + 4 - 4 = 9 - 4$, which results in $x = 5$. The division property may also be used to further simplify the equation and isolate the variable. By using these properties in combination, you can solve a wide range of linear equations.
Analyze how the addition property of equality can be used to solve equations involving decimals and fractions, and explain the importance of this property in the broader context of solving algebraic equations.
The addition property of equality is a fundamental tool in solving algebraic equations, and its application extends beyond just linear equations with integer coefficients. This property can also be used to solve equations involving decimals and fractions. For example, if you have the equation $\frac{1}{2}x + 1.5 = 4.25$, you can use the addition property to isolate the variable by adding the opposite of the constant term ($-1.5$) to both sides: $\frac{1}{2}x + 1.5 - 1.5 = 4.25 - 1.5$, which simplifies to $\frac{1}{2}x = 2.75$. Then, you can apply the division property to isolate the variable: $\frac{1}{2}x \div \frac{1}{2} = 2.75 \div \frac{1}{2}$, which results in $x = 5.5$. The versatility of the addition property in solving equations with various types of coefficients, including decimals and fractions, highlights its importance in the broader context of solving algebraic equations. By mastering the application of this property, you can develop a strong foundation for solving a wide range of mathematical problems.
The subtraction property of equality states that if two expressions are equal, subtracting the same number from both expressions will result in two new expressions that are also equal. This property is used in conjunction with the addition property to solve equations.
The division property of equality states that if two expressions are equal, dividing both expressions by the same non-zero number will result in two new expressions that are also equal. This property is used to solve equations by isolating a variable through division.
The process of finding the value of an unknown variable in an equation by applying the properties of equality, such as the addition and subtraction properties, to isolate the variable.