Question

Which of the following expressions is equivalent to \(7-2(y-1) ?\) (A) \(9-2 y\) (B) \(5-2 y\) (C) \(6-2 y\) (D) \(4+2 y\)

Short answer

The expression equivalent to \(7 - 2(y-1)\) is Option (A) \(9 - 2y\).

Step-by-step solution

Identify the Expression to Simplify

The given expression is \( 7 - 2(y-1) \). We will simplify this expression to find which of the options it is equivalent to.

Distribute the Negative Sign

Apply the distributive property to \(-2(y-1)\) by multiplying \(-2\) with each term inside the parentheses: \(-2(y-1) = -2 \cdot y + (-2) \cdot (-1) = -2y + 2\).

Combine Terms

Now, substitute the simplified form of \(-2(y-1)\) back into the original expression: \(7 - 2(y-1) = 7 + (-2y + 2) = 7 + 2 - 2y\).

Complete the Simplification

Combine the constants: \(7 + 2 = 9\). Thus, the simplified expression is \(9 - 2y\).

Find the Equivalent Expression

The simplified expression after all steps is \(9 - 2y\). Now, compare this with the provided options to find the equivalent expression: Option (A) \(9 - 2y\), Option (B) \(5 - 2y\), Option (C) \(6 - 2y\), Option (D) \(4 + 2y\). The expression \(9 - 2y\) is equivalent to Option (A).

Key concepts

Distributive Property

To successfully solve math equations, understanding the distributive property is key. It allows you to remove parentheses in an expression by distributing or multiplying the outside term with each term inside the parentheses.

In this exercise, you apply it to the expression \(-2(y-1)\) by multiplying \(-2\) with all terms inside the bracket. Here's how it goes:

• Multiply \(-2\) with \(y\) and get \(-2y\).
• Then, multiply \(-2\) with \(-1\) and remember that multiplying two negatives gives a positive, resulting in \(+2\).

The expression simplifies to \(-2y + 2\). This step reduces the complexity of the equation by managing parentheses, turning it into a more straightforward form. It's like peeling off layers to see the core of the problem beautifully simplified.

Combining Like Terms

Now that you have an expression without parentheses, it's time to combine like terms. This step is about simplifying the expression by grouping similar terms together.

In \(7 + 2 - 2y\), you identify that \(7\) and \(2\) are constants. Combining them together:

• Add \(7\) and \(2\) to get \(9\)

You are left with \(9-2y\).

Remember, like terms share the same variable raised to the same power. It's like seeing apples and oranges. By adding all the apples together, you can simplify and more easily work with your group of fruit. So, combining terms makes solving the equation easier and quicker.

Equivalent Expressions

After simplification using distributive property and combining terms, you end up with an expression that is easier to handle, \(9 - 2y\). But the big question is, how do you know it’s equivalent to one of the given options?

An equivalent expression means despite looking different initially, they hold the same value when calculated. Comparing the simplified expression \(9 - 2y\) with the options:

• Option (A) is \(9 - 2y\), which matches exactly.
• Other options either mismatch numerically or have different operations.

So, the expression \(7 - 2(y-1)\) is equivalent to \(9 - 2y\) as seen in Option (A). Having confidence in identifying equivalent expressions ensures you can simplify complex equations into manageable, recognizably equal terms!