Problem 56 - Beginning and Intermediate Algebra

Question

Let \(f(x)=-5 x+2\) and \(g(x)=x^{2}+7 x+2 .\) Find each of the following and simplify. \(k(x)=8 x-6 .\) Find \(x\) so that \(k(x)=0\)

Step-by-Step Solution

Answer

The value of x that satisfies k(x) = 0 is \(x = \frac{3}{4}\).

Step 1: Write down the given function k(x) and the equation k(x) = 0

We are given the function k(x) = 8x - 6. We need to find the value of x for which k(x) = 0. So, we need to solve the equation: \[8x - 6 = 0\]

Step 2: Solve for x

To find the value of x, we will isolate the variable x to one side of the equation. In the equation \(8x - 6 = 0\), we will first add 6 to both sides of the equation: \[8x = 6\] Next, we will divide both sides of the equation by 8: \[x = \frac{6}{8}\]

Step 3: Simplify the result

Now, we need to simplify the fraction \(\frac{6}{8}\) by dividing both the numerator and denominator by their greatest common divisor which is 2: \[x = \frac{6 ÷ 2}{8 ÷ 2} = \frac{3}{4}\] Thus, the value of x that satisfies k(x) = 0 is \(x = \frac{3}{4}\).

Key Concepts

Solving Linear Equations

When solving linear equations, our main goal is to find the value of the variable that makes the equation true. A linear equation typically looks like \( ax + b = 0 \). In our specific problem, we had the equation \( 8x - 6 = 0 \). The key steps to solve include:

• Isolate the variable: Move terms that do not contain the variable to the other side of the equation. This often involves addition or subtraction. For example, adding 6 to both sides in \( 8x - 6 = 0 \) gives us \( 8x = 6 \).
• Simplify the equation: Once the variable is isolated with a coefficient, divide or multiply to solve for the variable. Here we divided by 8 to solve \( x = \frac{6}{8} \).

Linear equations are straightforward once you master isolating the variable. This process involves simple arithmetic: addition, subtraction, multiplication, and division.

Function Evaluation

Function evaluation is about finding the output of a function given a certain input. A function like \( f(x) = -5x + 2 \) provides a specific rule telling us how to transform input \( x \) into a result or output \( f(x) \). To evaluate:

• Substitute the input value in place of \( x \) in the function.
• Perform any mathematical operations in the function expression.

For instance, if we wanted to evaluate \( f(1) \), we would substitute \( 1 \) for \( x \), yielding \( f(1) = -5(1) + 2 = -3 \).
Evaluating functions allows you to see how changes in input values affect the output. This is useful in understanding the behavior of mathematical models and equations.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common divisors other than 1. In our problem, the fraction \( \frac{6}{8} \) needed simplification.

• Identify the greatest common divisor (GCD) of both numbers involved.
• Divide both the numerator and the denominator by this GCD.

In this example, the GCD of 6 and 8 is 2. Dividing both the numerator and the denominator by 2 gives \( \frac{3}{4} \). Thus, \( \frac{3}{4} \) is the simplified version of \( \frac{6}{8} \).
Simplifying helps in making calculations easier and results clearer. It is a fundamental skill in algebra and necessary for clear and precise mathematical communication.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is the largest positive integer that evenly divides two or more numbers without leaving a remainder. It is an essential tool for simplifying fractions and ensuring numbers are in their simplest form.

• Find the factors: List out all the factors of the two numbers.
• Identify the largest common factor: Choose the largest factor that appears in both lists.

In the equation, numbers 6 and 8 needed simplification. The GCD of 6 and 8 is 2. So, dividing both by 2, we got \( \frac{3}{4} \).
Learning to find the GCD is vital for dealing with fractions and ratio problems. It simplifies complex calculations and helps achieve accurate results in math.