Solving Linear Equations Step By Step Fill Value For M And Calculate Y-Intercept

Introduction

In this article, we will walk through the steps of solving a linear equation, focusing on substituting the value of the slope ($m$) into the equation and then using a point's coordinates to find the $y$-intercept ($b$). This is a fundamental concept in algebra and is crucial for understanding linear functions and their graphs. We will start with the equation $y = mx + b$, which is the slope-intercept form of a linear equation, and use the given information to find the values of $m$ and $b$. By the end of this guide, you will have a clear understanding of how to substitute values into equations and solve for unknowns, a skill that is essential for various mathematical and real-world applications.

Understanding the Slope-Intercept Form

The slope-intercept form of a linear equation is expressed as $y = mx + b$, where:

• $y$ is the dependent variable.
• $m$ is the slope of the line, representing the rate of change of $y$ with respect to $x$.
• $x$ is the independent variable.
• $b$ is the $y$-intercept, which is the point where the line crosses the $y$-axis.

This form is particularly useful because it directly gives us two key pieces of information about the line: its slope and its $y$-intercept. Knowing these values makes it easy to graph the line and understand its behavior. The slope ($m$) tells us how steep the line is and whether it is increasing or decreasing. A positive slope means the line goes upwards from left to right, while a negative slope means it goes downwards. The $y$-intercept ($b$) tells us where the line starts on the $y$-axis.

Step 2 Filling the Value for $m$ into the Equation

Substituting the Slope

The first step in solving this problem is to substitute the given value of the slope ($m$) into the equation. Let's assume that the slope $m$ has been determined to be 3. So, we substitute $m = 3$ into the slope-intercept form, $y = mx + b$. This gives us:

$$y = 3x + b$$

This equation now represents a line with a slope of 3, but we still need to find the $y$-intercept ($b$). The $y$-intercept is the point where the line intersects the $y$-axis, and it is a crucial component in defining the line completely. Without knowing the $y$-intercept, we only know the direction and steepness of the line, but not its exact position on the coordinate plane.

Importance of Substitution

Substitution is a fundamental technique in algebra. It allows us to replace a variable with a known value, simplifying the equation and bringing us closer to a solution. In this case, substituting the slope helps us narrow down the possibilities for the line, making it easier to find the $y$-intercept. Without this substitution, we would have an equation with two unknowns ($m$ and $b$), making it impossible to solve directly.

Example Scenario

Imagine you are plotting a graph of a line. Knowing the slope helps you understand the angle at which the line rises or falls. If the slope is 3, for every 1 unit you move to the right on the $x$-axis, the line goes up 3 units on the $y$-axis. This gives you a sense of the line's steepness. However, without knowing the $y$-intercept, you don't know where to start drawing the line on the $y$-axis. Substituting the slope into the equation is the first step towards pinpointing the exact position and orientation of the line.

Step 3 Using Coordinates to Calculate the $y$-Intercept

Utilizing Given Points

After substituting the slope, we now have the equation $y = 3x + b$. The next step is to find the $y$-intercept ($b$). To do this, we need a point $(x, y)$ that lies on the line. Let's say we are given the coordinates $(9, 6)$. This means that when $x = 9$, $y = 6$. We can substitute these values into the equation to solve for $b$:

$$6 = 3(9) + b$$

This equation now has only one unknown, $b$, making it solvable. By substituting the coordinates of a point on the line, we have created a specific instance of the equation that allows us to isolate and calculate the $y$-intercept. The $y$-intercept is essential because it tells us where the line crosses the $y$-axis, giving us a fixed point to anchor the line.

Why Coordinates Matter

Each point on a line satisfies the equation of that line. By using the coordinates of a point, we are essentially saying that the equation must hold true for these specific values of $x$ and $y$. This is a powerful concept because it allows us to turn a general equation into a specific one that we can solve. Without knowing a point on the line, we would only have the slope to guide us, but not a fixed position.

Practical Application

Consider a real-world example where you are modeling the cost of a service based on the number of hours used. The slope represents the hourly rate, and the $y$-intercept represents the fixed cost. If you know the hourly rate (slope) and the total cost for a specific number of hours (a point), you can calculate the fixed cost ($y$-intercept). This is just one of many applications where knowing how to use coordinates to find the $y$-intercept can be invaluable.

Step 4 Solving for the $y$-Intercept

Isolating the Variable

Now that we have the equation $6 = 3(9) + b$, we can solve for $b$. First, we perform the multiplication:

$$6 = 27 + b$$

Next, to isolate $b$, we subtract 27 from both sides of the equation:

$$6 - 27 = b$$

$$-21 = b$$

So, the $y$-intercept ($b$) is -21. This means the line crosses the $y$-axis at the point (0, -21).

Importance of Solving for $b$

Solving for the $y$-intercept completes the process of defining the linear equation. Now that we know both the slope ($m = 3$) and the $y$-intercept ($b = -21$), we can write the complete equation of the line:

$$y = 3x - 21$$

This equation now fully describes the line, allowing us to graph it accurately, predict $y$ values for any given $x$, and understand the relationship between the variables.

Visualizing the Solution

Imagine plotting this line on a graph. You would start at the point (0, -21) on the $y$-axis. Since the slope is 3, for every 1 unit you move to the right on the $x$-axis, you would move up 3 units on the $y$-axis. This visualization helps to solidify the understanding of how the slope and $y$-intercept work together to define a line.

Real-World Relevance

In practical terms, the $y$-intercept often represents a starting value or a fixed cost. For example, in a business context, it might represent the initial investment required before any sales are made. Understanding how to solve for the $y$-intercept allows you to interpret and analyze such scenarios effectively.

Conclusion

In this article, we have covered the steps to fill the value for $m$ into the equation, use coordinates to calculate the $y$-intercept, and solve for the $y$-intercept. These are critical steps in understanding and working with linear equations. By mastering these techniques, you can confidently solve a wide range of problems in algebra and beyond. The ability to substitute values, use coordinates, and solve for unknowns is a fundamental skill that will serve you well in various mathematical and real-world contexts. Remember, practice is key to mastering these concepts, so continue to work through examples and apply these techniques to different problems. The more you practice, the more intuitive these steps will become, and the better you will understand the underlying principles of linear equations.