Mastering Slope-Intercept: Find Equations Equivalent To Y=(2/3)x-4
y = (2/3)x - 4, and it's going to be our benchmark throughout this exciting journey into the world of linear equations. This particular form, known as the slope-intercept form, is incredibly powerful because it gives us two crucial pieces of information about a line at a single glance: its slope and its y-intercept. Think of it like a line's DNA! In the general slope-intercept form, y = mx + b, the m stands for the slope, and the b stands for the y-intercept. For our specific target equation, y = (2/3)x - 4, we can immediately identify that the slope (m) is 2/3 and the y-intercept (b) is -4.What does this tell us? Well, the slope of 2/3 means that for every 3 units you move horizontally to the right on the graph (the "run"), the line will rise vertically by 2 units (the "rise"). It tells us the steepness and direction of our line. A positive slope like 2/3 indicates that the line is climbing upwards as you move from left to right across the graph. Imagine walking up a gentle hill—that's what a positive slope feels like! *This ratio of rise over run is fundamental to visualizing and understanding the trajectory of our line.*The y-intercept of -4 is equally important. This is the exact point where our line crosses the vertical y-axis. When x is 0, y is -4, so the line passes through the point (0, -4). This point serves as a critical anchor for drawing our line. Knowing both the slope and the y-intercept allows us to precisely graph the line without needing to plot multiple points. You simply start at the y-intercept and then use the slope to find other points on the line. *It's truly a marvel of mathematical efficiency!*To determine if any other equation is equivalent to our target y = (2/3)x - 4, we must ensure that it represents the exact same line. This means it must have both the identical slope (2/3) and the identical y-intercept (-4). If even one of these values differs, then the lines are not the same, and thus, the equations are not equivalent. It's not enough for them to just look similar; they must have the same underlying mathematical structure when expressed in the slope-intercept form. Our mission, then, is to systematically convert each given equation into the y = mx + b format and then perform a direct, clear comparison. This systematic approach eliminates guesswork and ensures accuracy. So, let's keep our target's characteristics—m = 2/3 and b = -4—firmly in mind as we evaluate each candidate equation. This will be our guiding light!y = (2/3)x - 4. Remember, our strategy is to transform each equation into the familiar y = mx + b form. This allows for a straightforward comparison of their slopes (m) and y-intercepts (b) with our target's values. Ready to become a detective of linear equations? Let's go!3x - 2y = 4. To convert this into slope-intercept form, our primary goal is to isolate y on one side of the equation. This is the cornerstone of our conversion process, ensuring that we can directly compare its m and b values with our target equation y = (2/3)x - 4. The initial step involves moving the x term to the other side of the equation. Since we have 3x on the left, we'll subtract 3x from both sides. This gives us -2y = -3x + 4. It’s crucial to pay close attention to the signs here, as a common mistake is to forget to carry over the negative with the 2y or to incorrectly sign the 3x after moving it.Once we have -2y = -3x + 4, the next logical step is to get y completely by itself. Currently, y is being multiplied by -2. To undo this multiplication, we must divide every single term on both sides of the equation by -2. Remember, whatever you do to one side, you must do to the other, and that applies to every term! So, dividing -2y by -2 leaves us with y. Dividing -3x by -2 results in (3/2)x. And finally, dividing +4 by -2 gives us -2. Putting it all together, our equation transforms into y = (3/2)x - 2.Now, the moment of truth! Let's compare this newly derived equation, y = (3/2)x - 2, with our target equation: y = (2/3)x - 4. The slope (m) of our target equation is 2/3. The slope (m) of 3x - 2y = 4 is 3/2.* These slopes are not the same.* The y-intercept (b) of our target equation is -4.* The y-intercept (b) of 3x - 2y = 4 is -2.* These y-intercepts are also not the same.Since both the slope and the y-intercept are different, we can confidently conclude that 3x - 2y = 4 is not equivalent to y = (2/3)x - 4. This clearly shows us that even minor differences in coefficients or constants can drastically change the line an equation represents. It's a fantastic example of why careful manipulation and comparison are essential in algebra.2x - 3y = 12. Just like before, our mission is to rewrite this in the elegant y = mx + b format. The first crucial step is to isolate the term containing y, so we need to move the 2x term from the left side to the right side of the equation. We achieve this by subtracting 2x from both sides. This action transforms our equation into -3y = -2x + 12. It's vital to ensure that you keep the negative sign with the 3y and correctly apply the negative sign to the 2x when moving it. Precision in these initial algebraic steps is what prevents errors down the line.With -3y = -2x + 12, our next objective is to get y all by itself. Currently, y is being multiplied by -3. To undo this operation, we must divide every single term on both sides of the equation by -3. This means dividing -3y by -3, dividing -2x by -3, and dividing +12 by -3. This step is often where students might forget to divide all terms, leading to incorrect slopes or y-intercepts.Performing these divisions, -3y / -3 simplifies to y. The term -2x / -3 becomes (2/3)x (notice how two negatives cancel out to form a positive!). Lastly, +12 / -3 simplifies to -4. Therefore, our second equation, when converted to slope-intercept form, becomes y = (2/3)x - 4.Now, for the exciting comparison! Let's stack this result against our target equation: y = (2/3)x - 4. The slope (m) of our target equation is 2/3. The slope (m) of 2x - 3y = 12 is 2/3. They match perfectly!. The y-intercept (b) of our target equation is -4. The y-intercept (b) of 2x - 3y = 12 is -4. *They also match perfectly!*Since both the slope and the y-intercept are identical to those of our target equation, we can definitively say that 2x - 3y = 12 is equivalent to y = (2/3)x - 4. This is a fantastic discovery, showing how different arrangements of numbers can represent the very same line! It reinforces the idea that an equation's form doesn't change its fundamental identity, much like a person can wear different outfits but remains the same individual.−4(2x − 3y) = −4(12). This one looks a little different because it involves multiplication on both sides, but don't let that intimidate you! Our first step is to simplify this expression, and there are a couple of ways to approach it. One effective method is to realize that both sides of the equation are being multiplied by -4. We can "undo" this by dividing both sides by -4. This will immediately simplify the equation and bring it closer to a standard form that we can then convert to slope-intercept.So, dividing −4(2x − 3y) by -4 leaves us with (2x − 3y). And dividing −4(12) by -4 leaves us with 12. Voila! The equation simplifies dramatically to 2x - 3y = 12. This transformation is a prime example of why understanding inverse operations is so powerful in algebra. It allowed us to bypass the distributive property initially, making the problem much quicker to solve.Now, astute readers might notice something familiar about 2x - 3y = 12. Wait a minute, haven't we seen this before? Indeed, we have! This is precisely the same equation as our second candidate that we just analyzed. Since we already confirmed that 2x - 3y = 12 is equivalent to y = (2/3)x - 4, then by extension, this equation, which simplifies directly to 2x - 3y = 12, must also be equivalent.To reiterate the conversion for completeness (though we already did it), let's quickly review:* Start with 2x - 3y = 12.* Subtract 2x from both sides: -3y = -2x + 12.* Divide all terms by -3: y = (-2/-3)x + (12/-3).* Simplify: y = (2/3)x - 4.Comparing y = (2/3)x - 4 to our target equation: y = (2/3)x - 4.* Slope (m) = 2/3 (Matches!).* Y-intercept (b) = -4 (Matches!).Therefore, −4(2x − 3y) = −4(12) is equivalent to y = (2/3)x - 4. This illustrates a key point: sometimes equations might look different on the surface, but after a bit of algebraic manipulation, they reveal themselves to be identical. It's like finding the same person wearing a different hat – still the same person underneath! This reinforces the idea that mathematical expressions can have various forms while retaining their core meaning.2(x + 6) = 3y. This one presents a slightly different initial structure, with y already on one side, though not fully isolated, and a distributive property waiting on the other. Our first order of business, as always, is to bring it closer to the y = mx + b form. We'll begin by applying the distributive property on the left side of the equation. This means multiplying 2 by x and 2 by 6.Performing the distribution, 2 * x becomes 2x, and 2 * 6 becomes 12. So, the left side transforms into 2x + 12. Our equation now reads 2x + 12 = 3y. This step is crucial for unwrapping the equation and preparing it for isolation.Now we have 2x + 12 = 3y. We want y to be by itself. Currently, y is being multiplied by 3. To isolate y, we need to divide every term on both sides of the equation by 3. This means 2x divided by 3, 12 divided by 3, and 3y divided by 3.Let's perform these divisions: 2x / 3 becomes (2/3)x. 12 / 3 simplifies to 4. And 3y / 3 simplifies to y. So, our equation becomes (2/3)x + 4 = y. To match the standard slope-intercept form (y = mx + b) more closely, we can simply rewrite this as y = (2/3)x + 4.Time for the comparison with our target equation: y = (2/3)x - 4. The slope (m) of our target equation is 2/3. The slope (m) of 2(x + 6) = 3y is 2/3. Yes, the slopes match! That's a good sign!. The y-intercept (b) of our target equation is -4. The y-intercept (b) of 2(x + 6) = 3y is +4. *Uh oh, they do not match.*Because the y-intercepts are different, even though the slopes are identical, we must conclude that 2(x + 6) = 3y is not equivalent to y = (2/3)x - 4. This is a fantastic illustration of parallel lines! Lines with the same slope but different y-intercepts are parallel; they run side-by-side forever, never intersecting. They have the same steepness and direction, but they start at different points on the y-axis. While related, they are distinct lines. Therefore, their equations are not equivalent.2x - 3y = 4. Our tried-and-true method remains the same: transform this equation into the y = mx + b format. The very first step involves isolating the y term, which means we need to move the 2x term from the left side to the right side of the equation. We accomplish this by subtracting 2x from both sides. This algebraic manipulation leaves us with -3y = -2x + 4. As always, double-check those signs to ensure accuracy; it's easy to make a small error here that cascades through the rest of the problem. Paying meticulous attention to detail at this stage is a hallmark of strong mathematical problem-solving.Now that we have -3y = -2x + 4, our next objective is to get y completely by itself. Since y is currently being multiplied by -3, the inverse operation is to divide every single term on both sides of the equation by -3. This includes -3y, -2x, and +4. This uniform division across all terms is critical to maintaining the balance and equality of the equation.Let's perform the divisions: -3y / -3 simplifies to y. The term -2x / -3 simplifies to (2/3)x. Remember, dividing a negative by a negative yields a positive, so the x term becomes positive. Lastly, +4 / -3 results in -4/3. So, after these operations, our equation takes the form y = (2/3)x - 4/3.Now, for the ultimate comparison with our target equation: y = (2/3)x - 4. The slope (m) of our target equation is 2/3. The slope (m) of 2x - 3y = 4 is 2/3. Fantastic, the slopes are a perfect match! This tells us these lines are at least parallel, perhaps even the same. The y-intercept (b) of our target equation is -4. The y-intercept (b) of 2x - 3y = 4 is -4/3. Unfortunately, here we hit a snag! The y-intercepts are different. While they are both negative, -4 is not the same as -4/3.Since the y-intercepts do not match, even though the slopes are identical, we must conclude that 2x - 3y = 4 is not equivalent to y = (2/3)x - 4. This again highlights the importance of matching both components (slope and y-intercept) for true equivalency. Just like in our previous example of 2(x + 6) = 3y, this equation also represents a line that is parallel to our target line but positioned differently on the coordinate plane. It has the same steepness, but it crosses the y-axis at a different point.y = mx + b and comparing them against our target equation, y = (2/3)x - 4, we can now deliver our final verdict. This systematic approach, focusing on isolating y and then directly comparing the slope (m) and the y-intercept (b), is undeniably the most reliable way to determine equivalency in linear equations. It strips away any superficial differences in how the equations are initially presented and reveals their true geometric identity. It’s like looking past the packaging to see the exact same product inside!Let's recap our findings for each candidate equation: Equation 1: 3x - 2y = 4 When converted, it became y = (3/2)x - 2.* Its slope (3/2) did not match our target's slope (2/3).* Its y-intercept (-2) did not match our target's y-intercept (-4).* Verdict: Not Equivalent. This line has a different steepness and crosses the y-axis at a different point.* Equation 2: 2x - 3y = 12* When converted, it beautifully transformed into y = (2/3)x - 4.* Its slope (2/3) perfectly matched our target's slope (2/3).* Its y-intercept (-4) perfectly matched our target's y-intercept (-4).* Verdict: Equivalent! This equation truly represents the exact same line as our target.* Equation 3: -4(2x - 3y) = -4(12)* This equation simplified directly to 2x - 3y = 12.* As we established with Equation 2, this form, when converted, yields y = (2/3)x - 4.* Both its slope (2/3) and y-intercept (-4) matched our target's values.* Verdict: Equivalent! A wonderful example of algebraic manipulation revealing an identical twin!* Equation 4: 2(x + 6) = 3y* Upon conversion, this equation became y = (2/3)x + 4.* Its slope (2/3) matched our target's slope (2/3).* However, its y-intercept (+4) did not match our target's y-intercept (-4).* Verdict: Not Equivalent. This is a parallel line – same direction, different starting point.* Equation 5: 2x - 3y = 4* After careful conversion, this equation transformed into y = (2/3)x - 4/3.* Its slope (2/3) matched our target's slope (2/3).* But its y-intercept (-4/3) did not match our target's y-intercept (-4).* Verdict: Not Equivalent. Another example of a parallel line with a different y-intercept.So, the equations that are truly equivalent to y = (2/3)x - 4 are:* 2x - 3y = 12* **-4(2x - 3y) = -4(12)**This exercise beautifully demonstrates the power of the slope-intercept form and the precision required in algebraic manipulation. It reminds us that appearances can be deceiving, and only a systematic approach can reveal the true relationships between mathematical expressions. Understanding these equivalencies is not just an academic exercise; it's a foundational skill for further study in mathematics, physics, engineering, and any field that relies on modeling relationships with lines. Keep practicing these conversions, and you'll master linear equations in no time!y = mx + b) and how to skillfully manipulate equations to reveal their true slopes and y-intercepts. Remember, two equations are only truly equivalent if they represent the exact same line, meaning they share both an identical slope and an identical y-intercept. This isn't just about finding matching numbers; it's about understanding the underlying geometric identity of the line itself. *This skill of transformation and comparison is a fundamental building block in mathematics, much like learning to read before you can write a novel.*The journey we took, converting each given equation step-by-step, highlights the importance of precision in every algebraic move, from distributing terms to dividing by negative numbers. Small errors in calculation can lead to entirely different slopes or y-intercepts, and thus, entirely different lines. *It's like baking a cake – even a tiny mismeasurement of an ingredient can change the whole outcome!By mastering these techniques, you're not just solving a problem; you're developing a deeper intuition for how linear relationships work, which is invaluable in countless real-world applications. From predicting trends in data to designing structures, understanding how lines behave is a superpower. Keep practicing, keep questioning, and keep exploring! Mathematics is a language, and the more fluent you become, the more of the world you'll understand.For those eager to deepen their understanding of linear equations and algebraic manipulation, here are some fantastic resources: Khan Academy offers a wealth of free tutorials and practice problems on algebra and linear equations. You can find their algebra resources at https://www.khanacademy.org/math/algebra * Desmos Graphing Calculator is an excellent interactive tool for visualizing lines and seeing how changes in slope and y-intercept affect their graphs. Explore it at https://www.desmos.com/calculator * Brilliant.org provides engaging lessons and problem-solving exercises across various math topics, including linear algebra. Check them out at **https://brilliant.org/courses/algebra-basics/**Keep learning and keep growing your mathematical mind!