Problem 100 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(97-108,\) determine whether the given number is a solution of the equation. $$-4 y+21=-7 y+15 ;-2$$

Step-by-Step Solution

Answer

-2 is a solution to the equation.

Step 1: Substitute the y-value into the equation

Substitute -2 for \(y\) in the equation: \[-4y+21 = -7y+15 \rightarrow -4(-2)+21 = -7(-2)+15\]

Step 2: Simplify both sides of the equation

Simplify both sides by performing the multiplication and addition:\[8+21 = 14+15 \rightarrow 29 = 29\]

Step 3: Verify equality

After simplifying, both sides of the equation are equal (29 = 29), verifying that \(y = -2\) is indeed a solution to the equation.

Key Concepts

Substitution Method

The substitution method is a powerful tool for solving algebraic equations and unlocking the mysteries of variables. In essence, this method involves replacing a variable with a given number to determine if the equation holds true. Think of it as a detective's technique for confirming an alibi in a mystery novel.
When you substitute a value into an equation, you're testing to see if the equation makes sense with that particular number. Imagine you're given an equation like \(-4y + 21 = -7y + 15\) and you need to find out if \(y = -2\) fits perfectly into this algebraic puzzle. You simply take the \(-2\), and place it gently in the y's position: \(-4(-2) + 21\) on one side and \(-7(-2) + 15\) on the other. Now, watch the magic unfold as you simplify the numbers - it's like pieces of a puzzle coming together.

Why use substitution?

• It's straightforward and methodical, making complex equations more manageable.
• It gives concrete results; either the equation balances, or it doesn't.
• It helps you understand how variables interact with numbers.

This method is perfect for beginners and experts alike as it strips down equations to their most basic form, revealing truths hidden within the numbers.

Algebraic Expressions

At the heart of algebra, algebraic expressions are the sentences that describe relationships between numbers and variables. They can be as simple as \(y + 5\) or as complex as \(4x^2 - 3xy + 7\). Each expression tells a story, and understanding that story is crucial to mastering algebra.
An expression like \(-4y + 21\) is like a code that needs deciphering. It says, 'Take a number, multiply it by -4, and then add 21.' When you're given an expression and a possible solution, you're essentially being asked, 'Does this number make my expression come true?' You test this by invoking the substitution method.

Components of Algebraic Expressions

• Coefficients: The numbers found next to variables, like the -4 in \(-4y\), give you the scaling factor of the variable.
• Variables: Symbols like y that represent unknown quantities. Changing these can change the entire story of the expression.
• Constants: Numbers without variables, like the 21, are the unchanging landmarks in the landscape of algebraic expressions.

By understanding each component and how they interact, you're not just crunching numbers; you're interpreting a language of numerical relationships that describe the world around us.

Verifying Solutions

Just as a chef tastes their dishes before serving, verifying solutions in algebra is a critical step in the problem-solving process. It's where you confirm your answers make sense, eliminating any doubts about the validity of your solution. Verifying isn't just about getting the right answer; it's about building confidence in your mathematical reasoning.
To verify a solution like \(y = -2\) in our equation \(-4y + 21 = -7y + 15\), you substitute \(-2\) into the equation and simplify. If both sides equal the same number, as they do with 29 = 29, you've just verified that \(y = -2\) is indeed the correct answer.

Why verification is vital:

• It's the ultimate check for accuracy; if the numbers don't match, it's back to the drawing board.
• It deepens your understanding, showing not only that an answer works but why it works.
• It's proof of your solving prowess; a verified solution is a proven solution.

Never skip verification; it's the seal of approval on your hard-earned answer, ensuring that you can step forward with confidence and clarity in your mathematical journey.