Equation Of A Line Passing Through (3, -2) With Slope 2/3

by James Vasile 58 views

Finding the equation of a line is a fundamental concept in algebra and coordinate geometry. In this article, we will explore how to determine the equation of a line when given a point it passes through and its slope. Specifically, we will focus on the scenario where the line passes through the point (3,−2)(3, -2) and has a slope of 23\frac{2}{3}. The goal is to express this equation in slope-intercept form, which is a standard way to represent linear equations. Guys, let's dive into the step-by-step process and make sure we fully understand each aspect.

Understanding Slope-Intercept Form

Before we jump into the problem, let's quickly recap the slope-intercept form of a linear equation. The slope-intercept form is expressed as:

y=mx+b{ y = mx + b }

where:

  • yy is the dependent variable (usually plotted on the vertical axis).
  • xx is the independent variable (usually plotted on the horizontal axis).
  • mm is the slope of the line, representing the rate of change of yy with respect to xx.
  • bb is the y-intercept, which is the point where the line crosses the y-axis.

The slope, denoted by mm, quantifies the steepness and direction of the line. It is calculated as the change in yy divided by the change in xx (rise over run). The y-intercept, bb, is the value of yy when xx is zero. Knowing these two parameters (mm and bb) allows us to uniquely define a line on the coordinate plane. Understanding slope-intercept form is crucial for visualizing and analyzing linear relationships.

Using the Point-Slope Form

When we have a point (x1,y1)(x_1, y_1) and the slope mm of a line, the point-slope form comes to our rescue. The point-slope form is given by:

y−y1=m(x−x1){ y - y_1 = m(x - x_1) }

This form is particularly useful because it allows us to directly plug in the given point and slope to create an equation. In our case, we have the point (3,−2)(3, -2) and the slope 23\frac{2}{3}. So, x1=3x_1 = 3, y1=−2y_1 = -2, and m=23m = \frac{2}{3}. Let's substitute these values into the point-slope form:

y−(−2)=23(x−3){ y - (-2) = \frac{2}{3}(x - 3) }

Simplifying, we get:

y+2=23(x−3){ y + 2 = \frac{2}{3}(x - 3) }

The point-slope form is a powerful tool for finding the equation of a line, especially when you have a specific point and the slope. It provides a direct and intuitive way to construct the equation, which can then be converted into other forms, such as the slope-intercept form, which we will do next. It's a handy method to keep in your mathematical toolkit.

Converting to Slope-Intercept Form

Our next step is to transform the equation from point-slope form to slope-intercept form (y=mx+by = mx + b). We have:

y+2=23(x−3){ y + 2 = \frac{2}{3}(x - 3) }

First, we distribute the 23\frac{2}{3} across the terms inside the parentheses:

y+2=23x−23(3){ y + 2 = \frac{2}{3}x - \frac{2}{3}(3) }

y+2=23x−2{ y + 2 = \frac{2}{3}x - 2 }

Now, to isolate yy, we subtract 2 from both sides of the equation:

y=23x−2−2{ y = \frac{2}{3}x - 2 - 2 }

y=23x−4{ y = \frac{2}{3}x - 4 }

This is the equation of the line in slope-intercept form. The slope mm is 23\frac{2}{3}, and the y-intercept bb is −4-4. This means the line rises 2 units for every 3 units it runs horizontally, and it crosses the y-axis at the point (0,−4)(0, -4). Converting to slope-intercept form not only gives us the equation in a standard format but also clearly reveals the slope and y-intercept, which are crucial for understanding the line's behavior on the coordinate plane. Guys, remember this process as it’s a key skill in algebra!

Final Equation and Interpretation

So, the equation of the line that passes through the point (3,−2)(3, -2) and has a slope of 23\frac{2}{3} in slope-intercept form is:

y=23x−4{ y = \frac{2}{3}x - 4 }

This equation tells us a few important things about the line. First, the slope of 23\frac{2}{3} indicates that for every 3 units we move to the right along the x-axis, the line goes up 2 units along the y-axis. This gives us the direction and steepness of the line. Second, the y-intercept of −4-4 tells us that the line crosses the y-axis at the point (0,−4)(0, -4). This is the point where the line intersects the vertical axis. Guys, visualizing this line on a graph can further enhance your understanding. The ability to interpret and derive such equations is essential for solving various problems in mathematics and real-world applications. Linear equations are fundamental tools in modeling and analyzing relationships between variables.

Alternative Methods and Verification

While we've used the point-slope form to find the equation, there are alternative methods you could use to verify your answer or solve the problem. For instance, you could plug the point (3,−2)(3, -2) directly into the slope-intercept form y=mx+by = mx + b and solve for bb. Since we know the slope m=23m = \frac{2}{3}, we have:

−2=23(3)+b{ -2 = \frac{2}{3}(3) + b }

−2=2+b{ -2 = 2 + b }

b=−4{ b = -4 }

This confirms our y-intercept. Another way to verify is to choose another point on the line and see if it satisfies the equation. For example, if we let x=6x = 6 in our equation y=23x−4y = \frac{2}{3}x - 4, we get:

y=23(6)−4{ y = \frac{2}{3}(6) - 4 }

y=4−4{ y = 4 - 4 }

y=0{ y = 0 }

So, the point (6,0)(6, 0) should also lie on the line. These alternative methods not only help in verifying your solution but also provide a deeper understanding of the concepts involved. It's always a good practice to use multiple approaches to reinforce your knowledge and ensure accuracy. Guys, trying different methods can make you a more confident problem-solver!

Common Mistakes to Avoid

When finding the equation of a line, there are a few common mistakes to watch out for. One frequent error is mixing up the coordinates when using the point-slope form. Make sure you correctly substitute x1x_1 and y1y_1 from the given point. Another mistake is incorrectly distributing the slope when converting from point-slope form to slope-intercept form. Remember to multiply the slope by both terms inside the parentheses. Additionally, be careful with the signs when simplifying the equation, especially when dealing with negative numbers. For example, subtracting a negative number is the same as adding the positive number.

Another common pitfall is not fully simplifying the equation to slope-intercept form. Make sure you isolate yy on one side of the equation. Always double-check your calculations and ensure you have the correct slope and y-intercept. By being mindful of these common errors, you can improve your accuracy and avoid unnecessary mistakes. Guys, practice makes perfect, so keep solving problems and learning from any mistakes you make!

Real-World Applications

The equation of a line has numerous real-world applications across various fields. In physics, it can describe the motion of an object moving at a constant velocity. In economics, it can represent cost functions or supply and demand curves. In computer graphics, lines are fundamental elements for drawing shapes and creating images. Linear equations are also used in statistics for linear regression, which helps to model the relationship between variables. For example, a line can be used to represent the relationship between hours studied and exam scores.

In everyday life, understanding linear equations can help you with budgeting, calculating distances, or even estimating travel times. The concept of slope and intercept is essential in many practical scenarios. For instance, if you are planning a road trip, you can use a linear equation to estimate the amount of fuel you'll need based on the distance and your car's fuel efficiency. The slope would represent the fuel consumption rate, and the y-intercept could represent the initial fuel level. Guys, the ability to apply mathematical concepts like this makes you a more effective problem-solver in all areas of life.

Conclusion

In summary, we have successfully found the equation of the line that passes through the point (3,−2)(3, -2) and has a slope of 23\frac{2}{3}. The equation in slope-intercept form is:

y=23x−4{ y = \frac{2}{3}x - 4 }

We began by understanding the slope-intercept form and then utilized the point-slope form to derive the equation. We then converted it to slope-intercept form and verified our solution using alternative methods. We also discussed common mistakes to avoid and explored real-world applications of linear equations. Guys, understanding these concepts is crucial for your mathematical journey. The ability to find and interpret equations of lines is a fundamental skill that will serve you well in various contexts. Keep practicing, and you'll become more confident in your problem-solving abilities! Remember, math is not just about numbers; it's about understanding the relationships and patterns that govern the world around us.