Problem 70 - Algebra 1

Question

CRITICAL THINKING Consider the equation \(6 x+8 y=k .\) What numbers could replace \(k\) so that the \(x\) -intercept and the \(y\) -intercept are both integers? Explain.

Step-by-Step Solution

Answer

The values that could replace k so that the x-intercept and the y-intercept are both integers are the common multiples of 6 and 8, such as 24, 48, 72, and any other multiple of these numbers.

Step 1: Identify the given equation

The given equation is \(6x + 8y = k\). This is a linear equation in two variables.

Step 2: Find the x-intercept

To find the x-intercept, set y = 0 in the equation and solve for x. The equation becomes \(6x + 8(0) = k\), which simplifies to \(6x = k\). Therefore, x would be \(x = \frac{k}{6}\). For x to be an integer, k must be a multiple of 6.

Step 3: Find the y-intercept

To find the y-intercept, set x = 0 in the equation and solve for y. The equation becomes \(6(0) + 8y = k\), which simplifies to \(8y = k\). Therefore, y would be \(y = \frac{k}{8}\). For y to be an integer, k must be a multiple of 8.

Step 4: Determine possible values of k

Based on step 2 and step 3, for both x and y to be integers, k must be a common multiple of 6 and 8. The common multiples of 6 and 8 are 24, 48, 72, and so on. So, the possible values for k are these common multiples.

Key Concepts

x-intercept

In linear equations, the x-intercept is the value of x when y equals zero. It is a critical point where the graph intersects the x-axis. To calculate the x-intercept of a line described by an equation like \[ 6x + 8y = k \]we substitute y with zero and solve for x. This leads us to the equation \[ 6x = k \]Then, we find \[ x = \frac{k}{6} \]For the x-intercept to be an integer, k must be a multiple of 6. Multiple of integers like 6 multiplies all integers to create numbers that maintain integer properties when divided back by 6.
Understanding x-intercepts helps simplify the analysis of linear equations and solve practical problems involving graphs where x-intercept represents a meaningful real-world situation.

y-intercept

The y-intercept is found where the graph of a linear equation crosses the y-axis, meaning x is zero. In our linear equation\[ 6x + 8y = k \]we find the y-intercept by inserting zero for x, which simplifies the equation to \[ 8y = k \]and further solving gives us \[ y = \frac{k}{8} \]The y-intercept will be an integer if k is a multiple of 8. This application of multiples lets us ensure y remains an integer value, maintaining clarity and precision in calculations that connect with real-world outcomes, such as starting points in data scenarios.
Grasping the essence of y-intercepts is crucial for students as they navigate through various mathematical challenges and applications.

multiple of integers

When dealing with equations and intercepts, understanding multiples is essential. Multiples of integers refer to numbers produced from multiplying an integer by another integer. For example, multiples of 6 are 6, 12, 18, etc.When finding both x- and y-intercepts of a linear equation, k must fulfill both multiple conditions, i.e., be a common multiple of 6 and 8.
The numbers that satisfy these conditions are called common multiples, such as 24, 48, and 72. This ensures both x and y in the equation\[ 6x + 8y = k \]are integers. Recognizing and applying multiples in math enhances accuracy and broad understanding in solving equations. Embracing these concepts promotes better critical thinking and problem-solving skills.

critical thinking

Critical thinking involves analyzing and evaluating an issue to make a judgment. In mathematics, particularly with linear equations, it's about understanding the steps necessary to arrive at a solution logically. When given\[ 6x + 8y = k \]it's essential to identify how the x- and y-intercepts are processed from the equation. To achieve \[ x = \frac{k}{6} \]and\[ y = \frac{k}{8} \]critical thinking allows us to deduce that k needs to be a common multiple of both 6 and 8 for x and y to both be integers.
This process entails recognizing patterns, evaluating the results, and ensuring all calculations actively contribute to finding valid solutions. Developing strong critical thinking skills in math aids students in approaching problems methodically and effectively.