Find The Equation Of A Line: Slope & Point

Jan 20, 2026 by Andrew McMorgan 43 views

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the awesome world of mathematics, specifically tackling a classic problem: finding the equation of a line when you've got the slope and a point it passes through. This is a fundamental skill, and once you get the hang of it, you'll see these types of problems everywhere, from your math class to real-world applications. So, let's break down how to solve this step-by-step, making sure we hit all the key points and understand why we're doing what we're doing. We've got a specific problem to work through: a line has a slope of $- rac{2}{3}$ and passes through the point $(-3,8)$ . Our mission is to find the equation of this line. Get ready to flex those math muscles!

Understanding the Building Blocks: Slope and Point

Before we jump into solving, let's get a solid grip on what we're dealing with. The slope of a line, often denoted by 'm', is basically a measure of its steepness and direction. It tells us how much the y-value changes for every one-unit increase in the x-value. A positive slope means the line goes upwards from left to right, while a negative slope means it goes downwards. A slope of $- rac{2}{3}$ means that for every 3 units we move to the right along the x-axis, the line drops 2 units in the y-direction. This ratio, $rac{ ext{rise}}{ ext{run}}$ , is crucial.

Now, the point $(-3,8)$ is simply a location on the coordinate plane that our line must pass through. The coordinates mean that when $x = -3$ , the corresponding $y$ value on the line is $8$ . This point is like an anchor for our line; it has to go through this specific spot. Combining the slope and a point gives us enough information to uniquely define a straight line. Think of it like this: you know how steep the road is (slope), and you know one specific address it passes by (point). With that, you can map out the entire road's route.

The Slope-Intercept Form: Our Best Friend

The most common and often the easiest form to work with when finding the equation of a line is the slope-intercept form. This equation looks like this: $y = mx + b$ . Here, 'm' is the slope (which we already know is $- rac{2}{3}$ in our problem), and 'b' is the y-intercept. The y-intercept is the point where the line crosses the y-axis. It's the value of y when $x = 0$ . Our goal is to find the value of 'b'. Once we have 'm' and 'b', we'll have the complete equation of the line.

So, we have our slope $m = - rac{2}{3}$ . We also have a point $(x,y) = (-3,8)$ that lies on the line. This means that when we plug $x = -3$ into the equation, the output must be $y = 8$ . This is the key insight that allows us to solve for 'b'. We'll substitute the known values of $x$ , $y$ , and $m$ into the slope-intercept equation and then solve the resulting algebraic equation for 'b'. It's a straightforward process that requires careful substitution and a bit of arithmetic. The beauty of the slope-intercept form is its directness; it explicitly shows you the slope and where the line hits the y-axis, making it super easy to graph and understand.

Step-by-Step Solution: Finding the Equation

Alright, let's get down to business and solve our problem. We know the slope $m = - rac{2}{3}$ and a point on the line $(x,y) = (-3,8)$ . We want to find the equation in the form $y = mx + b$ . Our first step is to substitute the known values of $m$ , $x$ , and $y$ into the slope-intercept equation:

$8 = (- rac{2}{3})(-3) + b$

Now, we need to simplify the right side of the equation. The multiplication $(- rac{2}{3})(-3)$ is a key step. Remember that multiplying two negative numbers results in a positive number. So, we have:

$(- rac{2}{3}) imes (-3) = rac{(-2) imes (-3)}{3} = rac{6}{3} = 2$

Substituting this back into our equation, we get:

$8 = 2 + b$

Our next step is to isolate 'b' to find its value. To do this, we subtract 2 from both sides of the equation:

$8 - 2 = 2 + b - 2$

$6 = b$

So, we've found that the y-intercept, 'b', is 6. This means our line crosses the y-axis at the point $(0,6)$ .

Assembling the Final Equation

Now that we have both the slope ( $m = - rac{2}{3}$ ) and the y-intercept ( $b = 6$ ), we can write the complete equation of the line by plugging these values back into the slope-intercept form, $y = mx + b$ .

$y = - rac{2}{3}x + 6$

And there you have it! This is the equation of the line that has a slope of $- rac{2}{3}$ and passes through the point $(-3,8)$ . We successfully used the given information to determine the missing piece (the y-intercept) and construct the final equation. It's a neat process that demonstrates how different pieces of information about a line can be combined to define it completely. This equation now perfectly describes all the points that lie on that specific line.

Alternative Method: Point-Slope Form

While the slope-intercept form is super useful, there's another powerful tool in our mathematical arsenal: the point-slope form. This form is particularly handy when you're given a point and the slope, just like in our problem. The point-slope form of a linear equation is given by: $y - y_1 = m(x - x_1)$ . Here, $m$ is the slope, and $(x_1, y_1)$ is a known point on the line.

Let's apply this to our problem. We have $m = - rac{2}{3}$ and the point $(x_1, y_1) = (-3,8)$ . We substitute these values directly into the point-slope formula:

$y - 8 = - rac{2}{3}(x - (-3))$

Notice the double negative for the x-coordinate: $x - (-3)$ becomes $x + 3$ . So, the equation is:

$y - 8 = - rac{2}{3}(x + 3)$

This equation is a valid equation for the line. However, it's not in the most commonly requested format, which is usually the slope-intercept form ( $y = mx + b$ ). To convert it, we need to do a bit of algebraic manipulation. First, distribute the slope $- rac{2}{3}$ to the terms inside the parentheses:

$y - 8 = - rac{2}{3}x + (- rac{2}{3})(3)$

Now, simplify the multiplication $(- rac{2}{3})(3)$ . This equals $-2$ . So the equation becomes:

$y - 8 = - rac{2}{3}x - 2$

Finally, to get the equation into slope-intercept form, we need to isolate $y$ . We do this by adding 8 to both sides of the equation:

$y - 8 + 8 = - rac{2}{3}x - 2 + 8$

$y = - rac{2}{3}x + 6$

As you can see, we arrive at the exact same answer using the point-slope form as we did with the slope-intercept form. Both methods are valid and lead to the correct equation. The point-slope form is a great starting point when you have a point and slope readily available, and it's often quicker to set up initially. Then, you just need that extra step to convert it to the desired slope-intercept form. It's all about having different tools for different jobs in your math toolkit!

Checking Your Work: Verification is Key!

So, we've found our equation: $y = - rac{2}{3}x + 6$ . But how do we know for sure it's correct? The best way is to verify it. We were given two pieces of information that our line must satisfy: its slope must be $- rac{2}{3}$ , and it must pass through the point $(-3,8)$ . Let's check both.

First, the slope. The equation $y = - rac{2}{3}x + 6$ is in slope-intercept form ( $y = mx + b$ ). By comparing the two, we can clearly see that $m = - rac{2}{3}$ . So, the slope condition is met. Easy peasy!

Second, let's check if the point $(-3,8)$ lies on this line. To do this, we substitute $x = -3$ into our equation and see if we get $y = 8$ . Remember, this is exactly what we did during our calculation to find 'b'.

Let's plug in $x = -3$ :

$y = - rac{2}{3}(-3) + 6$

We already calculated $(- rac{2}{3})(-3) = 2$ . So, the equation becomes:

$y = 2 + 6$

$y = 8$

Since substituting $x = -3$ gives us $y = 8$ , the point $(-3,8)$ indeed lies on the line represented by the equation $y = - rac{2}{3}x + 6$ . This confirms our solution is correct!

Why Verification Matters

Verification is a critical step in any problem-solving process, especially in math. It's not just about getting an answer; it's about ensuring that your answer is accurate and logically sound. In cases like finding the equation of a line, errors can creep in during calculation – maybe a sign flip or a multiplication mistake. By plugging your point back into the final equation or checking if the slope matches, you catch these potential blunders. It builds confidence in your answer and solidifies your understanding of the concepts. Plus, it's a great habit to get into for more complex problems down the line. Think of it as the final quality check before you declare your solution complete and correct. Never skip the verification step, guys!

Real-World Connections: Lines All Around Us

It might seem like just a math problem, but understanding how to find the equation of a line has tons of real-world applications. Lines represent constant rates of change, making them perfect for modeling many scenarios. For instance, if you're tracking the distance you travel on a road trip at a constant speed, that relationship is linear. The slope would be your speed, and the y-intercept could be any initial distance you've already covered.

Think about costs too. If a company has a fixed cost and then an additional cost per item produced, the total cost can be represented by a linear equation. The slope is the cost per item, and the y-intercept is the fixed cost. Businesses use this all the time for budgeting and forecasting. Even in physics, concepts like velocity and displacement often involve linear relationships. When you plot data points from an experiment, you might look for a line of best fit to understand the underlying trend, and that line has an equation!

Understanding these linear relationships helps us predict future outcomes, analyze trends, and make informed decisions. It's a fundamental concept that underpins many scientific and economic models. So, the next time you're solving for the equation of a line, remember that you're learning a skill that's genuinely useful and applicable far beyond the classroom. Keep practicing, and you'll be able to model and understand the world around you in a whole new way. Pretty cool, right?

Conclusion: Mastering Linear Equations

So, there you have it! We've successfully navigated the process of finding the equation of a line given its slope and a point. We explored the elegance of the slope-intercept form ( $y = mx + b$ ) and the utility of the point-slope form ( $y - y_1 = m(x - x_1)$ ). Both methods led us to the correct answer: $y = - rac{2}{3}x + 6$ . We also emphasized the crucial step of verification, ensuring our solution is accurate and reliable.

Remember the key steps: identify the slope ( $m$ ) and the given point $(x,y)$ . Then, either substitute these into $y = mx + b$ to solve for $b$ , or use the point-slope formula directly and convert to slope-intercept form. Whichever method you choose, the goal is to find that equation that perfectly describes the line's path. Mastering these linear equations is a huge step in your mathematical journey, opening doors to understanding more complex concepts and real-world applications. Keep practicing, keep questioning, and keep exploring the fascinating world of math. Until next time, stay sharp!

The correct option is:

$y=- rac{2}{3} x+6$