Solving Linear Equations And Finding Slopes A Comprehensive Guide

This article provides a detailed explanation of how to solve linear equations and find the slope of a line. We will walk through the steps involved in solving the equation $2 + \frac{x}{3} = 1 - 2x$ and finding the slope of the line $3x - 6y = 33$. These are fundamental concepts in algebra, and mastering them is crucial for further studies in mathematics and related fields. This comprehensive guide aims to provide a clear understanding of these concepts, making them accessible to learners of all levels.

329. Solving the Linear Equation: $2 + \frac{x}{3} = 1 - 2x$

To effectively solve the linear equation $2 + \frac{x}{3} = 1 - 2x$, we need to isolate the variable x on one side of the equation. This involves a series of algebraic manipulations to maintain the equality while simplifying the expression. Let's break down the process step-by-step to ensure clarity and understanding.

Step 1: Eliminate the Fraction

Fractions can often complicate the solving process, so the first step is to eliminate the fraction. In this equation, we have $\frac{x}{3}$. To get rid of the denominator, we multiply every term in the equation by 3. This ensures that the equation remains balanced, as we are performing the same operation on both sides.

$$3 \times \left(2 + \frac{x}{3}\right) = 3 \times (1 - 2x)$$

Distributing the 3 on both sides, we get:

$$3 \times 2 + 3 \times \frac{x}{3} = 3 \times 1 - 3 \times 2x$$

$$6 + x = 3 - 6x$$

Step 2: Gather the x Terms on One Side

Next, we want to group all the terms containing x on one side of the equation. To do this, we can add $6x$ to both sides. This will move the $-6x$ term from the right side to the left side.

$$6 + x + 6x = 3 - 6x + 6x$$

$$6 + 7x = 3$$

Step 3: Isolate the x Term

Now, we need to isolate the term with x. To do this, we subtract 6 from both sides of the equation. This will move the constant term from the left side to the right side.

$$6 + 7x - 6 = 3 - 6$$

$$7x = -3$$

Step 4: Solve for x

Finally, to solve for x, we divide both sides of the equation by 7. This will give us the value of x.

$$\frac{7x}{7} = \frac{-3}{7}$$

$$x = -\frac{3}{7}$$

Therefore, the solution to the equation $2 + \frac{x}{3} = 1 - 2x$ is $x = -\frac{3}{7}$.

Verification

To ensure our solution is correct, we can substitute $x = -\frac{3}{7}$ back into the original equation:

$$2 + \frac{-\frac{3}{7}}{3} = 1 - 2\left(-\frac{3}{7}\right)$$

$$2 - \frac{1}{7} = 1 + \frac{6}{7}$$

$$\frac{14}{7} - \frac{1}{7} = \frac{7}{7} + \frac{6}{7}$$

$$\frac{13}{7} = \frac{13}{7}$$

Since both sides of the equation are equal, our solution $x = -\frac{3}{7}$ is correct.

Conclusion

By following these steps, we have successfully solved the linear equation $2 + \frac{x}{3} = 1 - 2x$. The key to solving linear equations is to perform the same operations on both sides to maintain balance while isolating the variable. This methodical approach ensures accuracy and makes the process more manageable. Understanding these steps is fundamental for solving more complex algebraic problems.

330. Finding the Slope of the Line: $3x - 6y = 33$

Finding the slope of a line is a crucial concept in coordinate geometry. The slope represents the steepness and direction of a line. A positive slope indicates that the line is increasing (going upwards) as you move from left to right, while a negative slope indicates that the line is decreasing (going downwards). The slope is typically denoted by the letter m.

To determine the slope of the line given by the equation $3x - 6y = 33$, we need to rewrite the equation in slope-intercept form. The slope-intercept form of a linear equation is $y = mx + b$, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). Converting the given equation into this form will allow us to easily identify the slope.

Step 1: Rewrite the Equation in Slope-Intercept Form

To rewrite the equation $3x - 6y = 33$ in slope-intercept form ($y = mx + b$), we need to isolate y on one side of the equation. This involves a series of algebraic steps.

First, we subtract $3x$ from both sides of the equation:

$$3x - 6y - 3x = 33 - 3x$$

$$-6y = -3x + 33$$

Next, we divide both sides of the equation by -6 to solve for y:

$$\frac{-6y}{-6} = \frac{-3x + 33}{-6}$$

$$y = \frac{-3x}{-6} + \frac{33}{-6}$$

$$y = \frac{1}{2}x - \frac{11}{2}$$

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

Step 2: Identify the Slope

By comparing the equation $y = \frac{1}{2}x - \frac{11}{2}$ with the slope-intercept form $y = mx + b$, we can easily identify the slope. The coefficient of x is the slope m.

In this case, $m = \frac{1}{2}$.

Therefore, the slope of the line $3x - 6y = 33$ is $\frac{1}{2}$.

Understanding the Slope

The slope of $\frac{1}{2}$ tells us that for every 2 units we move horizontally (to the right) along the line, we move 1 unit vertically (upwards). This indicates a moderately steep, positive slope. The line is increasing as we move from left to right.

Alternative Method: Using Two Points on the Line

Another way to find the slope of a line is by using two points on the line. If we have two points $(x_1, y_1)$ and $(x_2, y_2)$, the slope m can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

To use this method, we need to find two points that satisfy the equation $3x - 6y = 33$.

1. Let's find the y-intercept by setting $x = 0$:

   $$3(0) - 6y = 33$$

   $$-6y = 33$$

   $$y = -\frac{33}{6} = -\frac{11}{2}$$

   So, one point is $(0, -\frac{11}{2})$.

2. Let's find another point by setting $x = 1$:

   $$3(1) - 6y = 33$$

   $$3 - 6y = 33$$

   $$-6y = 30$$

   $$y = -5$$

   So, another point is $(1, -5)$.

Now we can use the slope formula with $(x_1, y_1) = (0, -\frac{11}{2})$ and $(x_2, y_2) = (1, -5)$:

$$m = \frac{-5 - (-\frac{11}{2})}{1 - 0}$$

$$m = \frac{-5 + \frac{11}{2}}{1}$$

$$m = \frac{-\frac{10}{2} + \frac{11}{2}}{1}$$

$$m = \frac{\frac{1}{2}}{1}$$

$$m = \frac{1}{2}$$

We arrive at the same slope, $m = \frac{1}{2}$, confirming our earlier result.

Conclusion

Finding the slope of a line is a fundamental skill in algebra and geometry. By converting the equation to slope-intercept form or using the slope formula with two points, we can easily determine the slope. Understanding the slope provides valuable information about the line's direction and steepness, which is essential in various mathematical and real-world applications. The ability to manipulate equations and interpret their graphical representations is a cornerstone of mathematical proficiency.

Comprehensive Review

In summary, this article has provided a detailed walkthrough of solving the linear equation $2 + \frac{x}{3} = 1 - 2x$ and finding the slope of the line $3x - 6y = 33$. We covered the necessary algebraic steps, including eliminating fractions, isolating variables, and converting equations to slope-intercept form. These skills are essential for success in algebra and beyond.

By understanding these concepts and practicing regularly, learners can build a strong foundation in mathematics and develop the problem-solving skills necessary for more advanced topics. Linear equations and slopes are not just abstract mathematical concepts; they have practical applications in various fields, including physics, engineering, economics, and computer science. Therefore, mastering these concepts is an investment in future success.