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Chapter 4: Problem 31

Solve and check. $$-4-17=16 y-3(5 y+6)$$

Short Answer

Expert verified
Question: Solve the linear equation -4 - 17 = 16y - 3(5y + 6). Answer: y = -3

Step by step solution

01

Distribute the constant to the term with variable.

Apply the distributive property to the term \(-3(5 y+6)\): $$ -4-17=16 y-3(5 y+6)$$ $$ -4-17=16 y-15 y-18 $$
02

Combine like terms.

Combine the constants and the terms with variable y on both sides of the equation: $$ (-4-17)+18 = (16 y-15 y) $$ $$ -21+18 = 1y $$ $$ -3 = y $$
03

Check the solution.

Plug the value of y back into the original equation to ensure it is correct: $$ -4-17 \stackrel{?}{=} 16(-3)-3(5(-3)+6) $$ $$ -21 \stackrel{?}{=} -48+81 $$ $$ -21 = -21 $$ The solution is correct. The final solution is: $$ y = -3 $$

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Key Concepts

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

Distributive Property
The distributive property is a fundamental algebraic principle that simplifies expressions and equations. It allows us to multiply a single term across terms inside a parenthesis. In mathematical terms, it is expressed as:
  • \( a(b + c) = ab + ac \)
In the exercise given, we applied the distributive property to the term \(-3(5y + 6)\). This means multiplying \(-3\) with each term inside the parenthesis:
  • \(-3 \cdot 5y = -15y\)
  • \(-3 \cdot 6 = -18\)
So, we arrived at the expression \(-15y - 18\), which can then be combined with terms outside the parenthesis for further simplification. The distributive property is crucial for transforming expressions and making them easier to handle, setting the stage for the next steps.
Combine Like Terms
Combining like terms is an essential skill in solving equations. It involves simplifying expressions by grouping and adding or subtracting terms that have the same variables and exponents. Let's break it down:
  • "Like terms" are terms whose variables (and their exponents) are identical. In the expression \(16y - 15y\), both terms feature the variable \(y\) to the same power.
  • Similarly, constants like \(-4\), \(-17\), and \(18\) are also considered like terms as they do not have any variable part.
In the original problem, we start by simplifying the equation. On the left side :
  • \(-4 - 17 + 18 = -3\)
On the right side :
  • \(16y - 15y = 1y\)
This simplification using the combination of like terms leads us to a clear, concise equation: \(-3 = y\). Highly useful in reducing complexity, this step is key to solving equations effectively.
Check the Solution
After solving an equation, it's important to verify your solution to ensure accuracy. This is done by substituting the solution back into the original equation.
  • Begin by rewriting the original equation and substituting the variable with the solution. Here, \(y = -3\).
In our exercise, we substitute \(-3\) for \(y\) in the equation:
  • \(-4 - 17 = 16(-3) - 3(5(-3) + 6)\)
  • This simplifies to \(-21 = -48 + 81\)
Conducting these calculations should leave you with the same result on both sides of the equation, confirming the accuracy of the solution:
  • \(-21 = -21\)
Checking your solution is an essential step in problem-solving. It gives confidence that your solution is correct and reinforces your understanding of the problem-solving process.

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