Explain why you need Just the addition property of equality in the multiplication property of equally and not also the subtraction property of equally in the division property of equally to solve this question in this lesson give an example to illustrate your reasoning
To understand why only the addition property of equality is needed in the multiplication property of equality, and not the subtraction property of equality in the division property of equality, let's first review the properties of equality.
The properties of equality are a set of rules that help us manipulate equations and prove statements about their relationships. They include the addition property, subtraction property, multiplication property, division property, reflexive property, symmetric property, and transitive property.
- Addition Property of Equality: If we add the same value to both sides of an equation, the equation remains true. For example, if we have the equation x + 2 = 6, we can add 3 to both sides, giving us x + 2 + 3 = 6 + 3, which simplifies to x + 5 = 9.
- Subtraction Property of Equality: If we subtract the same value from both sides of an equation, the equation remains true. For example, if we start with x + 5 = 9, we can subtract 2 from both sides, resulting in x + 5 - 2 = 9 - 2, which simplifies to x + 3 = 7.
- Multiplication Property of Equality: If we multiply both sides of an equation by the same value, the equation remains true. For example, if we have the equation 3x = 12, we can multiply both sides by 2, giving us 2 * (3x) = 2 * 12, which simplifies to 6x = 24.
- Division Property of Equality: If we divide both sides of an equation by the same non-zero value, the equation remains true. For example, if we start with 6x = 24, we can divide both sides by 6, resulting in (6x) / 6 = 24 / 6, which simplifies to x = 4.
Now, let's consider a question to illustrate why only the addition property of equality is needed in the multiplication property, and not the subtraction property in division.
Example question: Solve the equation 4x + 7 = 31.
To solve this equation, we want to isolate the variable x on one side of the equation. Our goal is to perform a series of operations to transform the equation while ensuring its equality remains intact.
First, we can subtract 7 from both sides of the equation using the subtraction property of equality:
(4x + 7) - 7 = 31 - 7.
This simplifies to:
4x = 24.
At this point, we want to isolate x. To do that, we can use the multiplication property of equality. If we multiply both sides of the equation by ¼ (or divide both sides by 4), we can simplify the equation further:
4x * ¼ = 24 * ¼.
This yields:
x = 6.
As you can see, we only needed the addition property of equality to manipulate the equation in the beginning. The subtraction property of equality is not necessary in this case because the desired operation was to multiply by a fraction, not subtract.