How can you determine which property of equality is needed to isolate a variable and solve a given equation? Explain.
To determine which property of equality is needed to isolate a variable and solve a given equation, you need to consider the operations performed on the variable in the equation.
The properties of equality that can be used to isolate a variable and solve an equation are:
1. Addition Property of Equality: This property allows you to add the same value to both sides of an equation. It is useful when the variable is being subtracted on one side, and you want to isolate it by canceling out the subtraction.
For example, in the equation x - 5 = 10, we can apply the addition property of equality by adding 5 to both sides: (x - 5) + 5 = 10 + 5, which simplifies to x = 15.
2. Subtraction Property of Equality: This property allows you to subtract the same value from both sides of an equation. It is helpful when the variable is being added on one side, and you want to isolate it by cancelling out the addition.
For instance, in the equation x + 7 = 20, we can apply the subtraction property of equality by subtracting 7 from both sides: (x + 7) - 7 = 20 - 7, which simplifies to x = 13.
3. Multiplication Property of Equality: This property allows you to multiply both sides of an equation by the same value. It is useful when the variable is divided or multiplied by a value, and you want to cancel out that operation.
For example, in the equation 2x = 18, we can apply the multiplication property of equality by multiplying both sides by 1/2: (1/2)(2x) = (1/2)(18), which simplifies to x = 9.
4. Division Property of Equality: This property allows you to divide both sides of an equation by the same non-zero value. It is helpful when the variable is multiplied by a value, and you want to undo the multiplication.
For instance, in the equation 3y = 30, we can apply the division property of equality by dividing both sides by 3: (3y)/3 = 30/3, which simplifies to y = 10.
By examining the operations performed on the variable, you can determine which property of equality to use in order to isolate the variable and solve the equation.
To determine which property of equality is needed to isolate a variable and solve a given equation, you need to understand the different properties of equality and analyze the equation carefully. Here are the four main properties of equality:
1. Reflexive property: This property states that any quantity is equal to itself. For example, for any value of x, x = x.
2. Symmetric property: This property states that if two quantities are equal to each other, then they can be interchanged without affecting the truth of the equation. For example, if a = b, then b = a.
3. Transitive property: This property states that if quantity A equals quantity B, and quantity B equals quantity C, then quantity A equals quantity C. For example, if a = b and b = c, then a = c.
4. Substitution property: This property allows you to replace a quantity with an equal value. For example, if a = b, then you can substitute b for a or vice versa.
To isolate a variable and solve an equation, you need to perform a series of mathematical operations to get the variable alone on one side of the equation. The property you utilize depends on the specific equation and the operations needed to isolate the variable.
Here is a step-by-step process to determine which property of equality is needed:
1. Identify the variable: Determine which letter or symbol represents the variable in the equation. Typically, the variable is denoted by x, but it can be any other letter or symbol.
2. Analyze the equation: Review the equation and identify any constants, coefficients, or terms that involve the variable.
3. Perform necessary operations: Look for patterns or rules to isolate the variable. Use properties such as addition, subtraction, multiplication, division, and combining like terms to manipulate the equation.
4. Apply the appropriate property: Once you identify the operation you need to perform, apply the relevant property of equality to make a valid transformation.
5. Simplify and solve: Continue manipulating the equation until the variable is isolated on one side, and other terms are on the other side. Simplify the equation further if required and solve for the variable.
It is essential to choose the appropriate property of equality at each step to maintain the equality and ensure the equation remains true. By thoroughly understanding the properties and analyzing the equation, you can effectively isolate the variable and solve the given equation.
to undo addition, use the subtraction property
to undo multiplication, use the division property
and so on