Transforming 5x - Y = 10 Into Slope-Intercept Form A Step-by-Step Guide

Introduction: Mastering Linear Equations

Linear equations form the bedrock of algebra and are fundamental to various mathematical and scientific disciplines. Understanding how to manipulate and interpret these equations is crucial for problem-solving and analytical reasoning. One of the most versatile and insightful forms of a linear equation is the slope-intercept form, expressed as y = mx + b, where m represents the slope and b denotes the y-intercept. This form provides a clear visualization of the line's characteristics, making it easier to graph, analyze, and compare different linear relationships. In this comprehensive guide, we will delve into the process of transforming the linear equation 5x - y = 10 into slope-intercept form, elucidating each step with detailed explanations and examples. By mastering this transformation, you will gain a deeper understanding of linear equations and their applications.

Before we dive into the transformation, let's first grasp the significance of the slope-intercept form. The slope, m, quantifies the steepness and direction of a line. A positive slope indicates an upward trend, while a negative slope signifies a downward trend. The magnitude of the slope reflects the rate of change; a larger slope implies a steeper line. The y-intercept, b, represents the point where the line intersects the y-axis. This point provides a crucial reference for plotting the line and understanding its position on the coordinate plane. By expressing a linear equation in slope-intercept form, we gain immediate access to these key parameters, facilitating a wide range of applications.

Transforming an equation into slope-intercept form involves isolating the y variable on one side of the equation. This process typically involves algebraic manipulations such as adding, subtracting, multiplying, or dividing both sides of the equation by the same value. The goal is to rearrange the equation so that it conforms to the y = mx + b format. In the case of 5x - y = 10, we will employ a series of steps to achieve this transformation. First, we will subtract 5x from both sides of the equation to begin isolating the y term. This will give us a new equation that is closer to the desired slope-intercept form. Then, we'll need to address the negative sign in front of the y term, which we will do by multiplying both sides of the equation by -1. This will leave us with y isolated and the equation in the familiar y = mx + b format. Each of these steps will be explained in detail in the sections that follow, ensuring that you have a clear understanding of the entire transformation process.

Step-by-Step Transformation of 5x - y = 10 into Slope-Intercept Form

1. Isolating the y-term: Subtracting 5x from Both Sides

To initiate the transformation of the linear equation 5x - y = 10 into slope-intercept form, our primary objective is to isolate the y term on one side of the equation. This involves strategically manipulating the equation while maintaining its balance. The first step in this process is to eliminate the 5x term from the left side of the equation. To achieve this, we perform the inverse operation, which is subtracting 5x from both sides of the equation. This ensures that the equation remains balanced, as any operation performed on one side must be mirrored on the other side.

Subtracting 5x from both sides of the equation can be written as follows:

5x - y - 5x = 10 - 5x

This operation effectively cancels out the 5x term on the left side, leaving us with:

-y = 10 - 5x

Now, the equation appears closer to the slope-intercept form, but we still have a negative sign associated with the y term. This negative sign needs to be addressed before we can fully express the equation in y = mx + b format. The next step will involve eliminating this negative sign, which we will accomplish by multiplying both sides of the equation by -1. This manipulation will ensure that y is positive and that the equation is in the desired form for identifying the slope and y-intercept.

2. Eliminating the Negative Sign: Multiplying by -1

Following the isolation of the -y term, the next crucial step in transforming the equation 5x - y = 10 into slope-intercept form is to eliminate the negative sign associated with y. To accomplish this, we employ the fundamental algebraic principle of multiplying both sides of the equation by the same value. In this case, we will multiply both sides by -1. This operation effectively changes the sign of each term in the equation, ensuring that y becomes positive.

Multiplying both sides of the equation -y = 10 - 5x by -1, we get:

(-1) * (-y) = (-1) * (10 - 5x)

This simplifies to:

y = -10 + 5x

Now, the equation has a positive y term, but it is not yet in the standard slope-intercept form, which is y = mx + b. The terms on the right side of the equation are in the reverse order. To align with the standard form, we need to rearrange the terms so that the 5x term comes before the -10. This rearrangement will not change the mathematical meaning of the equation but will make it easier to identify the slope and y-intercept at a glance.

3. Rearranging the Equation: Achieving Slope-Intercept Form

With the negative sign eliminated, the equation now reads y = -10 + 5x. While mathematically correct, this form deviates from the standard slope-intercept form, y = mx + b, where the x term precedes the constant term. To fully transform the equation into slope-intercept form, we need to rearrange the terms on the right side of the equation. This rearrangement is a simple matter of changing the order of the terms, taking care to maintain their respective signs.

By rearranging the terms, we obtain:

y = 5x - 10

This equation is now in the quintessential slope-intercept form. We can clearly identify the slope, m, as 5 and the y-intercept, b, as -10. This transformation has provided us with a clear and concise representation of the linear relationship, allowing for easy interpretation and application. The slope of 5 indicates that for every one-unit increase in x, y increases by 5 units. The y-intercept of -10 signifies that the line intersects the y-axis at the point (0, -10). This information is invaluable for graphing the line, analyzing its behavior, and comparing it with other linear equations.

Interpreting the Slope-Intercept Form: y = 5x - 10

Now that we have successfully transformed the equation 5x - y = 10 into slope-intercept form, y = 5x - 10, it is essential to understand the implications of this form. The slope-intercept form provides a wealth of information about the linear relationship, allowing us to easily visualize the line's characteristics and behavior. The two key parameters that we can extract from this form are the slope (m) and the y-intercept (b).

Unveiling the Slope: m = 5

The slope, denoted by m, is the coefficient of the x term in the slope-intercept form. In our equation, y = 5x - 10, the slope is 5. The slope quantifies the steepness and direction of the line. A positive slope indicates that the line rises as we move from left to right, while a negative slope indicates that the line falls. The magnitude of the slope reflects the rate of change; a larger slope implies a steeper line.

A slope of 5 signifies that for every one-unit increase in x, the value of y increases by 5 units. This can be visualized as