Find The Y-Intercept: $(-3x-6)(-5x-15)=y$

Hey guys, welcome back to Plastik Magazine! Today, we're diving deep into the world of mathematics, specifically tackling a problem that might look a little intimidating at first glance, but trust me, it's totally manageable. We're on a mission to determine the y-intercept of the equation $(-3x-6)(-5x-15)=y$. You know, that sweet spot where the graph of our equation decides to say hello to the y-axis. It's all about finding that specific point where $x$ is zero, and $y$ gets to do its thing. So, buckle up, grab your favorite beverage, and let's break this down together. We'll explore what a y-intercept really is, how to find it algebraically, and apply that knowledge to our given equation. Get ready to boost your math game and impress your friends with your newfound intercept-finding skills!

Understanding the Y-Intercept: More Than Just a Point

Alright, let's get our heads around what a y-intercept actually is, because it's a fundamental concept in algebra and graphing. Basically, whenever you have an equation that describes a relationship between $x$ and $y$, the y-intercept is the specific coordinate point where the graph of that equation crosses the y-axis. Now, the defining characteristic of any point on the y-axis is that its x-coordinate is always, without fail, zero. Think about it: the y-axis is the vertical line where $x=0$. So, if a point is on that line, its $x$ value must be 0. This little nugget of information is our golden ticket to finding the y-intercept algebraically. All we need to do is substitute $x=0$ into our equation and solve for $y$. The resulting coordinate pair $(0, y)$ is our y-intercept. It tells us the starting value of $y$ when $x$ begins at zero. This concept is super important in understanding linear equations, where the y-intercept often represents an initial value or a starting point before any change occurs. For instance, in a linear equation representing cost over time, the y-intercept might be the fixed startup cost, regardless of how many hours you use the service. It's a crucial piece of the puzzle for visualizing and interpreting mathematical relationships. So, remember: when in doubt, set x to zero!

The Algebraic Approach to Finding the Y-Intercept

Now that we've got a solid grip on what the y-intercept is, let's talk about the nitty-gritty of how to find it using algebra. The method is beautifully straightforward, and it directly stems from our definition: the y-intercept occurs when $x=0$. So, the algebraic approach is as simple as plugging in zero for every instance of $x$ in your equation and then solving for the resulting value of $y$. Once you have that $y$ value, you pair it with the $x=0$ you used, and voilà! You have your y-intercept in the form $(0, y)$. It’s like a secret code where $x=0$ unlocks the value of $y$ at that specific crossing point. This technique works for pretty much any type of equation, whether it's linear, quadratic, or even more complex functions. The underlying principle remains the same. For our specific problem, the equation is given in a factored form: $y = (-3x - 6)(-5x - 15)$. To find the y-intercept, we substitute $x=0$ into this equation. This gives us $y = (-3(0) - 6)(-5(0) - 15)$. Simplifying this, we get $y = (0 - 6)(0 - 15)$, which further simplifies to $y = (-6)(-15)$. Multiplying these two negative numbers together results in a positive number. And $-6$ times $-15$ equals $90$. So, when $x=0$, $y=90$. Therefore, the y-intercept is the point $(0, 90)$. This algebraic method is your most reliable tool for pinpointing the y-intercept without needing to graph the equation, which can be super handy when dealing with complex functions or when you just want a quick and accurate answer. It’s all about substitution and simplification, guys!

Applying the Method to Our Equation: Step-by-Step

Let's roll up our sleeves and apply the algebraic method to our specific equation: $y = (-3x - 6)(-5x - 15)$. Remember, our goal is to find the y-intercept, which happens when $x=0$. So, the first step is to substitute $x=0$ into the equation. This looks like:

$y = (-3(0) - 6)(-5(0) - 15)$

Now, we simplify the terms inside the parentheses. Multiplying any number by zero results in zero, so:

$y = (0 - 6)(0 - 15)$

This simplifies the equation further to:

$y = (-6)(-15)$

Finally, we perform the multiplication. When you multiply two negative numbers, the result is positive. So, $(-6) imes (-15)$ equals $90$.

$y = 90$

Since the y-intercept is defined as the point where $x=0$, and we found that when $x=0$, $y=90$, our y-intercept is the coordinate pair $(0, 90)$. This means the graph of the equation crosses the y-axis at the point where $y$ has a value of $90$. It’s as simple as that! No need to expand the whole quadratic unless you want to verify, but for finding the y-intercept, this substitution method is a direct route. Pretty neat, huh?

Expanding the Equation (Optional Verification)

Sometimes, you might see the equation presented in its expanded, standard quadratic form, like $ax^2 + bx + c = y$. While our current equation $y = (-3x - 6)(-5x - 15)$ is in factored form, we can expand it to see what the standard form looks like and confirm our y-intercept. This step isn't strictly necessary to find the y-intercept (as we've already done it efficiently!), but it's a good way to double-check our work and understand the full structure of the equation. To expand, we'll use the distributive property (often called FOIL for binomials: First, Outer, Inner, Last):

1. First terms: $(-3x) imes (-5x) = 15x^2$
2. Outer terms: $(-3x) imes (-15) = 45x$
3. Inner terms: $(-6) imes (-5x) = 30x$
4. Last terms: $(-6) imes (-15) = 90$

Now, we combine these results:

$y = 15x^2 + 45x + 30x + 90$

Combine the like terms (the $x$ terms):

$y = 15x^2 + 75x + 90$

So, the expanded form of our equation is $y = 15x^2 + 75x + 90$. In any quadratic equation in the standard form $y = ax^2 + bx + c$, the constant term, $c$, is always the y-intercept. In our expanded equation, $c = 90$. This confirms that our y-intercept is indeed $(0, 90)$. This expansion method reinforces the idea that the constant term in a polynomial represents its value when the variable (in this case, $x$) is zero. It’s a great way to connect different forms of equations and verify your findings. Pretty cool how math offers so many pathways to the same answer, right?

Analyzing the Options: Which One Is It?

We've done the hard math work, guys, and figured out that the y-intercept for the equation $y = (-3x - 6)(-5x - 15)$ is the point $(0, 90)$. Now, let's look at the multiple-choice options provided and see which one matches our result. Remember, the y-intercept must have an x-coordinate of 0.

• A. $(0,-2)$ and $(0,-3)$: These are potential y-intercepts, but our calculation gave us a single, specific point.
• B. $(90,0)$: This point has a y-coordinate of 0, making it an x-intercept, not a y-intercept.
• C. $(2,0)$ and $(-3,0)$: These are x-intercepts, where $y=0$. Definitely not our y-intercept.
• D. $(0,90)$: This point has an x-coordinate of 0 and a y-coordinate of 90. This matches our calculated y-intercept perfectly!
• E. $(-2,0)$ and $(-3,0)$: These are also x-intercepts.
• F. $(0,-90)$: This point has an x-coordinate of 0, but the y-coordinate is negative 90, which doesn't match our positive 90.

So, after comparing our determined y-intercept with the given options, it's clear that Option D: $(0,90)$ is the correct answer. It's always a good practice to check your options against your calculated answer, especially in a test scenario. This ensures you haven't made any silly mistakes and that you've selected the right choice.

Conclusion: Mastering the Y-Intercept

And there you have it, mathletes! We've successfully navigated the process to determine the y-intercept of the equation $y = (-3x - 6)(-5x - 15)$. By understanding that the y-intercept is simply the point where the graph crosses the y-axis (meaning $x=0$), we were able to substitute $x=0$ into the equation. This yielded $y = (-6)(-15)$, which ultimately gave us $y = 90$. Thus, the y-intercept is the coordinate pair $(0, 90)$. We also took a little detour to expand the equation into its standard quadratic form, $y = 15x^2 + 75x + 90$, confirming that the constant term, $c=90$, is indeed the y-intercept. This provides a solid verification of our initial method. Remember, guys, the key takeaway here is the power of substitution: when you need to find the y-intercept, just set $x=0$ and solve for $y$. It’s a fundamental skill that opens the door to understanding graphs and functions more deeply. Keep practicing these concepts, and you'll be an algebra whiz in no time! Thanks for joining us here at Plastik Magazine for this math breakdown. Catch you in the next one!