Equivalent Equations Of A Line A Comprehensive Guide
Hey guys! Today, we're diving deep into the fascinating world of linear equations. We've got a line that bravely passes through the points (-4, 10) and (-1, 5), and it's currently flaunting the equation y = -5/3(x - 2). But, like a chameleon, this line can rock different outfits (equations) while still being the same awesome line. Our mission, should we choose to accept it, is to unmask three other equations that are secretly the same line in disguise. Buckle up, because we're about to embark on a mathematical adventure that's as thrilling as it is enlightening!
Unraveling the Mystery Equation: y = -5/3(x - 2)
Our starting point, the given equation y = -5/3(x - 2), is like a coded message. To truly understand it, we need to break it down and see what it tells us about the line. This equation is presented in what's known as the point-slope form. It's a super handy way to represent a line when you know a point on the line and the slope (how steep the line is). Let's dissect this equation piece by piece.
First up, the slope. In our equation, the slope is -5/3. Remember, the slope is the rate at which the line rises or falls as you move from left to right. A negative slope, like ours, means the line slopes downward. For every 3 units you move to the right on the graph, the line drops 5 units. Think of it like skiing downhill – exhilarating, but you gotta watch your speed!
Next, we have the (x - 2) part. This tells us about a point that lies on the line. The point-slope form is generally written as y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope. So, in our case, x1 = 2. But what about y1? Hold that thought! The equation is given as y = -5/3(x - 2), which is essentially y - 0 = -5/3(x - 2). This means y1 = 0. So, we've unearthed a point on our line: (2, 0). This point is like a secret rendezvous spot on our linear journey.
Now, let's confirm that the points (-4, 10) and (-1, 5) indeed lie on this line. This is crucial because if these points don't fit the equation, we're dealing with a different line altogether! To do this, we'll plug in the x and y values of each point into our equation and see if the equation holds true. It's like a mathematical fingerprint test – if the fingerprints match, we've got our culprit!
For the point (-4, 10): 10 = -5/3(-4 - 2) becomes 10 = -5/3(-6), which simplifies to 10 = 10. Bingo! The point (-4, 10) is definitely a resident of our line.
For the point (-1, 5): 5 = -5/3(-1 - 2) becomes 5 = -5/3(-3), which simplifies to 5 = 5. Double bingo! The point (-1, 5) is also part of our linear crew.
So, we've confirmed that the given equation, y = -5/3(x - 2), represents a line with a slope of -5/3 and passes through the points (-4, 10) and (-1, 5). We've successfully deciphered the coded message and now we're ready to find its aliases.
The Quest for Equivalent Equations: Unmasking the Imposters
Now comes the fun part: our quest to find three other equations that secretly represent the same line. It's like being a detective, sifting through clues and using our mathematical prowess to unmask the imposters. Remember, equivalent equations are just different ways of writing the same relationship between x and y. They might look different on the surface, but underneath, they're all telling the same story.
To find these equivalent equations, we'll use a few key strategies. Think of them as our detective tools: the distributive property, combining like terms, and rearranging equations. These tools will allow us to transform our original equation into different forms while preserving its core meaning. It's like taking a lump of clay and molding it into different shapes – it's still the same clay, just in a new form.
Tool #1: The Distributive Property – Our Mathematical Multiplier
The distributive property is like a mathematical multiplier, allowing us to spread a factor across terms within parentheses. It's the secret ingredient for expanding expressions and simplifying equations. In our case, we'll use it to get rid of the parentheses in our original equation, y = -5/3(x - 2). Think of it as unleashing the power within the parentheses!
Applying the distributive property, we multiply -5/3 by both x and -2: y = (-5/3) * x + (-5/3) * (-2). This simplifies to y = -5/3x + 10/3. And just like that, we've transformed our equation into a new form! This equation, y = -5/3x + 10/3, is a prime suspect for being one of our equivalent equations. It's in slope-intercept form (y = mx + b), which makes it easy to see the slope (-5/3) and the y-intercept (10/3).
Tool #2: Combining Like Terms – Our Neatness Ninja
Combining like terms is like being a neatness ninja, tidying up equations by grouping similar terms together. It helps us simplify equations and make them easier to work with. While we didn't need this tool in our initial transformation, it's a valuable technique to keep in our detective toolkit for more complex scenarios. Think of it as the Marie Kondo method for equations – sparking joy by decluttering and organizing!
Tool #3: Rearranging Equations – Our Mathematical Magician
Rearranging equations is like being a mathematical magician, manipulating equations to isolate variables or put them in a specific form. It's a powerful technique for solving equations and finding equivalent forms. In our quest, we'll use it to transform our equations into different formats, like standard form (Ax + By = C). Think of it as the art of mathematical alchemy – turning equations into gold!
Let's take our newly transformed equation, y = -5/3x + 10/3, and work our magic. To get rid of the fraction, we can multiply both sides of the equation by 3: 3y = -5x + 10. Now, let's move the -5x to the left side to get the equation in standard form: 5x + 3y = 10. This equation, 5x + 3y = 10, is another strong contender for being an equivalent equation. It's a whole new look for our line, but it's still the same line underneath.
Spotting the Suspects: Analyzing the Options
Now that we've mastered our detective tools and transformed our original equation, it's time to put our skills to the test. We have a lineup of potential equivalent equations, and it's our job to identify the three that truly match our line. It's like a mathematical whodunit, and we're the star detectives!
Let's revisit the options presented in the problem. We need to carefully analyze each one and see if it's a match for our line. We'll use our knowledge of slopes, intercepts, and equivalent forms to make our deductions. Think of it as a mathematical game of Clue, where we piece together the evidence to reveal the truth.
Option 1: y = -5/3x - 2
This equation looks suspiciously similar to our slope-intercept form, y = -5/3x + 10/3. The slopes match (-5/3), which is a good sign. But the y-intercepts are different: -2 versus 10/3. This means the line represented by this equation would cross the y-axis at a different point than our original line. So, this equation is an imposter! It's trying to blend in, but the y-intercept gives it away.
Option 2: y = -5/3x + 10/3
Ah, this equation looks familiar! We actually derived this equation ourselves using the distributive property. It's in slope-intercept form, and both the slope (-5/3) and the y-intercept (10/3) match our line. This is definitely one of our equivalent equations! We've unmasked one of the suspects.
Option 3: 3y = -5x + 10
This equation is intriguing. It's not in slope-intercept form, but it looks like it could be related to our standard form equation. To be sure, let's rearrange it to slope-intercept form by dividing both sides by 3: y = (-5/3)x + 10/3. Lo and behold, it's the same equation as Option 2! This is another equivalent equation, just disguised in a different form. We're on a roll!
Option 4: 5x + 3y = 10
This equation is in standard form, and it rings a bell. We actually derived this equation by rearranging our slope-intercept form. It's a perfect match! This is our third equivalent equation. We've successfully identified all three suspects.
Option 5: y = 5/3x + 10/3
This equation is a tricky one. The y-intercept matches our line (10/3), but the slope is different (5/3 versus -5/3). A different slope means the line will have a different steepness and direction. This equation is definitely not an equivalent equation. It's trying to fool us with the matching y-intercept, but the slope betrays its true identity.
The Grand Reveal: The Equivalent Equations Unveiled
After our meticulous investigation, we've successfully unmasked the three equivalent equations that represent the line passing through (-4, 10) and (-1, 5). It's like solving a complex puzzle, where each piece of information fits together to reveal the bigger picture. The thrill of the chase and the satisfaction of finding the solution – that's the beauty of mathematics!
The equivalent equations are:
- y = -5/3x + 10/3
- 3y = -5x + 10
- 5x + 3y = 10
These equations might look different, but they all tell the same story about our line. They're like different languages expressing the same idea. We've shown how we can transform one equation into another using our mathematical tools, highlighting the flexibility and interconnectedness of mathematical concepts. It's like a mathematical symphony, where different instruments (equations) play together in harmony to create a beautiful melody (the line).
Final Thoughts: The Power of Equivalent Forms
This exploration into equivalent equations demonstrates a crucial concept in mathematics: the power of representation. A single mathematical relationship can be expressed in countless ways, each with its own advantages and insights. Understanding equivalent forms allows us to choose the representation that best suits our needs, making problem-solving more efficient and elegant. It's like having a toolbox full of different tools – the right tool for the right job.
So, the next time you encounter an equation, remember that it's just one way of telling a story. There are countless other ways to tell the same story, and exploring these different perspectives can deepen your understanding and unlock new possibilities. Keep exploring, keep questioning, and keep embracing the beauty and power of mathematics! You've got this!