Finding The Y-Intercept A Comprehensive Guide

Jul 9, 2025 by ADMIN 46 views

Decoding the $y$-intercept of a Line: A Comprehensive Guide

Understanding the $y$ -intercept is crucial in grasping linear equations and their graphical representations. The $y$ -intercept is the point where a line crosses the $y$ -axis on a coordinate plane. It's the value of $y$ when $x$ is zero. This article delves deep into how to find the $y$ -intercept, focusing on the specific problem of a line passing through the point $(5, -6)$ with a slope of $-\frac{1}{7}$ . We will explore the underlying concepts, step-by-step solutions, and the significance of the $y$ -intercept in various mathematical and real-world contexts.

Understanding the Slope-Intercept Form

To effectively tackle this problem, it's essential to understand the slope-intercept form of a linear equation. The slope-intercept form is expressed as:

$y = mx + b$

Where:

$y$ is the dependent variable (the vertical axis).
$x$ is the independent variable (the horizontal axis).
$m$ represents the slope of the line, indicating its steepness and direction.
$b$ is the $y$ -intercept, the point where the line intersects the $y$ -axis (when $x = 0$ ).

The slope, denoted by $m$ , quantifies the rate of change of $y$ with respect to $x$ . It's calculated as the rise (vertical change) over the run (horizontal change) between any two points on the line. A positive slope indicates an upward trend, while a negative slope indicates a downward trend. The $y$ -intercept, $b$ , provides a crucial anchor point for graphing the line, as it directly tells us where the line crosses the vertical axis.

In our problem, we are given the slope $m = - rac{1}{7}$ and a point $(5, -6)$ that lies on the line. Our goal is to find the value of $b$ , the $y$ -intercept. To do this, we'll use the given information to first determine the equation of the line and then isolate the $b$ term.

Step-by-Step Solution: Finding the $y$ -intercept

1. Utilizing the Point-Slope Form

Since we have a point and a slope, the most efficient way to start is by using the point-slope form of a linear equation:

$y - y_1 = m(x - x_1)$

Where:

$(x_1, y_1)$ is the given point on the line.
$m$ is the slope.

In our case, $(x_1, y_1) = (5, -6)$ and $m = - rac{1}{7}$ . Plugging these values into the point-slope form, we get:

$y - (-6) = - rac{1}{7}(x - 5)$

This equation represents the line in point-slope form, which is a valuable intermediate step towards finding the slope-intercept form.

2. Simplifying to Slope-Intercept Form

Now, we need to simplify the equation and convert it to the slope-intercept form ( $y = mx + b$ ). First, simplify the left side:

$y + 6 = - rac{1}{7}(x - 5)$

Next, distribute the $- rac{1}{7}$ on the right side:

$y + 6 = - rac{1}{7}x + rac{5}{7}$

To isolate $y$ , subtract 6 from both sides of the equation:

$y = - rac{1}{7}x + rac{5}{7} - 6$

Now, we need to combine the constant terms. To do this, we'll express 6 as a fraction with a denominator of 7:

$6 = rac{6 imes 7}{7} = rac{42}{7}$

Substitute this back into the equation:

$y = - rac{1}{7}x + rac{5}{7} - rac{42}{7}$

Combine the fractions:

$y = - rac{1}{7}x + rac{5 - 42}{7}$

$y = - rac{1}{7}x - rac{37}{7}$

Now, the equation is in slope-intercept form: $y = mx + b$ .

3. Identifying the $y$ -intercept

From the slope-intercept form $y = - rac{1}{7}x - rac{37}{7}$ , we can directly identify the $y$ -intercept, $b$ . The $y$ -intercept is the constant term in the equation.

Therefore, the $y$ -intercept is:

$b = - rac{37}{7}$

So, the line intersects the $y$ -axis at the point $(0, - rac{37}{7})$ .

Why is the $y$ -intercept Important?

The $y$ -intercept is more than just a point on a graph; it holds significant meaning in various mathematical and real-world contexts. Here are a few reasons why it's important:

Starting Point: In the context of linear functions, the $y$ -intercept represents the initial value or starting point. For example, if the equation represents the cost of a service, the $y$ -intercept might represent the fixed fee or initial cost before any usage.
Graphing Lines: The $y$ -intercept, along with the slope, provides a straightforward way to graph a line. Plot the $y$ -intercept on the $y$ -axis, then use the slope (rise over run) to find other points on the line.
Problem Solving: In many word problems, the $y$ -intercept provides a crucial piece of information for setting up and solving equations. It often represents a constant value or a fixed condition.
Data Interpretation: In data analysis, the $y$ -intercept can help interpret the relationship between variables. For example, in a scatter plot, the $y$ -intercept might represent the predicted value of the dependent variable when the independent variable is zero.

Understanding the $y$ -intercept enhances your ability to analyze linear relationships, interpret data, and solve a wide range of problems.

Common Mistakes to Avoid

When finding the $y$ -intercept, there are a few common mistakes to watch out for:

Incorrectly Applying the Point-Slope Form: Ensure you substitute the values of $x_1$ , $y_1$ , and $m$ correctly into the point-slope form equation.
Algebraic Errors: Pay close attention to signs and operations when simplifying the equation. A small error in algebra can lead to an incorrect $y$ -intercept.
Misinterpreting the Slope-Intercept Form: Remember that the $y$ -intercept is the constant term ( $b$ ) in the $y = mx + b$ equation. Don't confuse it with the slope ( $m$ ).
Forgetting to Simplify: Always simplify the equation to slope-intercept form before identifying the $y$ -intercept. This ensures you're reading the correct value.
Not Double-Checking: Before finalizing your answer, review your steps and make sure your solution makes sense in the context of the problem.

By avoiding these common pitfalls, you can confidently and accurately find the $y$ -intercept of any linear equation.

Practice Problems

To solidify your understanding, try solving these practice problems:

Find the $y$ -intercept of the line passing through the point $(2, 5)$ with a slope of $3$ .
Determine the $y$ -intercept of the line passing through the point $(-1, -4)$ with a slope of $- rac{2}{3}$ .
What is the $y$ -intercept of the line that passes through the point $(0, 7)$ with a slope of $-1$ ?
A line has a slope of $\frac{1}{2}$ and passes through the point $(4, -2)$ . Find its $y$ -intercept.
The line passes through $(-3, 1)$ and has a slope of $2$ . What is its $y$ -intercept?

Working through these problems will reinforce your skills and build your confidence in finding $y$ -intercepts.

Real-World Applications

The concept of the $y$ -intercept extends beyond textbook problems and has practical applications in real-world scenarios. Here are a few examples:

Cost Analysis: In business, the $y$ -intercept of a cost function might represent the fixed costs, such as rent or insurance, that a company incurs regardless of production volume.
Distance-Time Graphs: In physics, the $y$ -intercept of a distance-time graph could represent the initial position of an object before it starts moving.
Financial Planning: In personal finance, the $y$ -intercept of a savings or investment graph might represent the initial investment amount.
Temperature Conversion: The formula to convert Celsius to Fahrenheit ( $F = rac{9}{5}C + 32$ ) has a $y$ -intercept of 32, which is the Fahrenheit equivalent of 0 degrees Celsius.
Sales and Commissions: In sales, the $y$ -intercept might represent a base salary before any commissions are earned.

By recognizing the $y$ -intercept in these real-world contexts, you can gain valuable insights and make informed decisions.

Conclusion

In summary, finding the $y$ -intercept of a line involves understanding the slope-intercept form ( $y = mx + b$ ) and using the point-slope form ( $y - y_1 = m(x - x_1)$ ) when given a point and a slope. By carefully applying these concepts and avoiding common mistakes, you can confidently determine the $y$ -intercept, which holds significant meaning in mathematics and various real-world applications. In the specific case of the line passing through the point $(5, -6)$ with a slope of $- rac{1}{7}$ , we found the $y$ -intercept to be $- rac{37}{7}$ . Mastering this skill enhances your problem-solving abilities and deepens your understanding of linear relationships.

By practicing and applying these concepts, you'll strengthen your understanding of linear equations and their applications, ultimately improving your mathematical skills and problem-solving abilities. Remember, the $y$ -intercept is a key component in understanding the behavior and characteristics of linear functions.