Problem 58 - Introductory Algebra for College Students

Question

Will help you prepare for the material covered in the next section. a. If \(y=\frac{k}{x},\) find the value of \(k\) using \(x=8\) and \(y=12\) b. Substitute the value for \(k\) into \(y=\frac{k}{x}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=3\)

Step-by-Step Solution

Answer

The constant of variation \(k\) is 96, the resulting equation is \(y = \frac{96}{x}\), and when \(x = 3\), the value of \(y\) is 32.

Step 1: Compute the value of \(k\)

Given that \(y = \frac{k}{x}\), we can find \(k\) by simply rearranging the formula to \(k = y \cdot x\). Using the given values \(y = 12\) and \(x = 8\), we substitute these into the equation to find \(k = 12 \cdot 8\).

Step 2: Calculate \(k\)

Using the known rules of multiplication, the result of \(12 \cdot 8\) results in \(k = 96\). Hence, the value of \(k\) in the equation \(y = \frac{k}{x}\) is 96.

Step 3: Substitute the value of \(k\) into the equation

Now that we have found the value of \(k\), we can substitute \(k = 96\) into the original equation, resulting in our new direct variation equation \(y = \frac{96}{x}\).

Step 4: Compute \(y\) when \(x = 3\)

To find the value of \(y\), we substitute \(x = 3\) into the equation \(y = \frac{96}{x}\). This gives us an equation \(y = \frac{96}{3}\).

Step 5: Calculate \(y\)

By dividing 96 by 3, we get \(y = 32\). Therefore, when \(x = 3\), \(y\) equals to 32.

Key Concepts

Solving for a Constant

Understanding how to solve for a constant is integral in algebra. A constant is a fixed and well-defined number. In the context of direct and inverse variation equations, it represents the proportionality or variation factor.

Practicing With Direct Variation

Consider an equation of the form \(y = \frac{k}{x}\), where \(k\) is the constant to be determined. If two values are provided, such as \(x=8\) and \(y=12\), you can find the constant by multiplying these values together (based on the rearranged formula \(k = y \times x\)). It is crucial to accurately identify the constant as it defines the relationship between variables for all subsequent values.

Substitution Method in Algebra

The substitution method is a vital tool for solving equations in algebra. It involves replacing a variable with its equivalent value to simplify the equation and solve for unknown quantities.

Applying Substitution

Once the constant \(k\) is determined, as in \(k = 96\), it's substituted back into the original equation in place of \(k\). This process simplifies the equation to \(y = \frac{96}{x}\), making it ready for use in finding new values of \(y\) for different \(x\) inputs. Careful substitution guarantees an accurate relationship between the variables.

Inverse Variation

Inverse variation is a type of relationship where one variable increases as the other decreases. The product of the two variables remains constant.

Recognizing Inverse Relationships

Equations like \(y = \frac{k}{x}\) demonstrate inverse variation. If you increase \(x\), \(y\) decreases so that their product is always the constant \(k\). It's pivotal to notice that in inverse variation, unlike direct variation, there is an indirect relationship; as one quantity grows, the other reduces proportionately.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They are foundational elements in algebra that describe relationships between variables.

Constructing Expressions

The equation from the exercise, \(y = \frac{96}{x}\), is an algebraic expression representing an inverse variation. Mastering how to interpret and manipulate these expressions is key to solving algebraic problems, allowing one to spell out the nature of relationships quantitatively and to predict the behavior of one variable in response to changes in another.