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Chapter 3: Problem 11

Determine whether the given equation is an identity or a contradiction. $$7-3(y-2)=y+4(5-y)-7$$

Short Answer

Expert verified
The equation is an identity.

Step by step solution

01

Distribute and Simplify Both Sides

First, distribute and simplify both sides of the equation. For the left side: \(7 - 3(y - 2) = 7 - 3y + 6\) Combine like terms: \(13 - 3y\).For the right side: \(y + 4(5 - y) - 7 = y + 20 - 4y - 7\)Combine like terms: \(y + 20 - 4y - 7 = 13 - 3y\).
02

Compare the Simplified Equations

After simplifying, both sides of the equation are the same: Left Side: \(13 - 3y\)Right Side: \(13 - 3y\).Since both sides are identical for any value of \(y\), the equation is an identity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

identity equations
An identity equation is an equation that holds true for all values of the variable involved. In other words, no matter what number you substitute for the variable, both sides of the equation will always remain equal. For example, when we solve the equation given in the exercise, we found that both sides simplified to the same expression: \(13 - 3y\). This means that no matter what value is plugged in for \(y\), the equation will always hold true, making it an identity equation.
simplifying expressions
Simplifying expressions is a fundamental step in solving algebraic equations. This process involves combining like terms and performing arithmetic operations to make the equation easier to solve. In the exercise, we simplified both sides of the equation:
  • For the left side: Distributing gives \(7 - 3(y - 2) = 7 - 3y + 6\). Combining like terms results in \(13 - 3y\).
  • For the right side: Distributing gives \(y + 4(5 - y) - 7 = y + 20 - 4y - 7\). Combining like terms results in \(13 - 3y\).
By simplifying expressions, we ensure the equation is easier to understand and solve.
distributive property
The distributive property is an essential algebraic rule that allows you to multiply a single term by each term inside a set of parentheses. It can be written as: \(a(b + c) = ab + ac\). In the exercise, we applied the distributive property to simplify both sides of the equation:
  • For the left side: \( -3(y - 2) = -3y + 6\)
  • For the right side: \(4(5 - y) = 20 - 4y\)
Using the distributive property helps in breaking down and simplifying complex expressions.
like terms
Like terms are terms that have the same variable raised to the same power. They can be combined to simplify an expression. In the given exercise, we combined like terms on each side of the equation to simplify it further:
  • For the left side: \(-3y\) and constant terms \(7 + 6\)
  • For the right side: \(y\) and \(-4y\) with constant terms \(20 - 7\)
Combining like terms is crucial for reducing equations to their simplest form, making it easier to solve them.

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Most popular questions from this chapter

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Translate the given phrase or sentence algebraically. \(h\) is no less than \(8 .\)

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