Solving System Of Equations: 4x=3y-2, -20x+15y=10

Hey math enthusiasts! Today, we're diving into a fascinating system of equations. We'll break down the steps to solve it, making sure everyone, from math newbies to seasoned pros, can follow along. Let's get started and unravel this mathematical puzzle together!

Understanding the Equations

Before we jump into solving, let's take a good look at the equations we're dealing with. We have two equations here, and both are linear equations. That means when you graph them, they'll form straight lines. The solutions to a system of linear equations are the points where these lines intersect. So, our goal is to find out if these lines intersect, and if they do, where!

Breaking Down the First Equation: 4x = 3y - 2

The first equation, 4x = 3y - 2, might look a bit jumbled at first glance, but it’s simpler than you think. In this equation, we see that the variable 'x' is multiplied by 4, and this product is equal to 3 times the variable 'y' minus 2. To make things clearer, we can rearrange this equation into the more familiar slope-intercept form (y = mx + b), where 'm' represents the slope of the line and 'b' represents the y-intercept. This form helps us visualize and understand the line's behavior on a graph. To rearrange, we'll first add 2 to both sides of the equation, resulting in 4x + 2 = 3y. Next, we'll divide the entire equation by 3 to isolate 'y'. This gives us y = (4/3)x + (2/3). Now, we can clearly see that the slope of this line is 4/3 and the y-intercept is 2/3. Understanding the slope and y-intercept provides key insights into the line's direction and position on the coordinate plane.

Analyzing the Second Equation: -20x + 15y = 10

Now, let's shift our focus to the second equation: -20x + 15y = 10. This equation also presents a linear relationship between 'x' and 'y', but it's expressed in a slightly different format. Here, we have 'x' multiplied by -20 and 'y' multiplied by 15, and these terms combine to equal 10. Similar to the first equation, we can rearrange this one into the slope-intercept form (y = mx + b) to make it easier to compare and analyze. This rearrangement will help us identify the slope and y-intercept of this second line. To do this, we'll first add 20x to both sides of the equation, which gives us 15y = 20x + 10. Then, we'll divide the entire equation by 15 to isolate 'y'. This results in y = (20/15)x + (10/15), which simplifies to y = (4/3)x + (2/3). Notice anything familiar? This equation, in its simplified form, reveals some crucial information about the line it represents.

Methods to Solve Systems of Equations

Okay, guys, there are a few ways we can tackle systems of equations. Think of them as different tools in our math toolbox. We've got substitution, elimination, and even graphing! Each method has its own strengths, and the best one to use often depends on the specific equations we're dealing with. For this particular system, we'll explore a couple of these methods to show you how they work and what they reveal about the solution.

The Substitution Method

The substitution method is a handy technique for solving systems of equations, especially when one equation can easily be solved for one variable in terms of the other. It involves isolating one variable in one equation and then substituting that expression into the other equation. This effectively reduces the system to a single equation with a single variable, which can then be solved directly. Once you've found the value of one variable, you can plug it back into either of the original equations to find the value of the other variable. It’s like a mathematical game of “plug and chug,” where we substitute one expression for another to simplify the problem and find our solution.

The Elimination Method

The elimination method, also known as the addition method, is another powerful tool for solving systems of equations. It works by manipulating the equations so that when you add them together, one of the variables cancels out. This is achieved by multiplying one or both equations by a constant so that the coefficients of one variable are opposites (e.g., 5x and -5x). Once you add the equations, you're left with a single equation in one variable, which you can solve. Then, like the substitution method, you can plug the value you found back into one of the original equations to solve for the other variable. This method is particularly useful when the equations are in standard form (Ax + By = C), as it makes the elimination process more straightforward. The key to this method is strategic manipulation to make the cancellation happen, turning a complex system into a simple equation.

Solving the System

Let's roll up our sleeves and get to solving! We'll use both the substitution and elimination methods to show you how they work in practice. This way, you can see which one clicks best with you. Remember, the goal is to find the values of 'x' and 'y' that satisfy both equations simultaneously.

Applying the Elimination Method

Remember how we rearranged both equations into slope-intercept form? Well, here's where that pays off big time. Notice that both equations, 4x = 3y - 2 and -20x + 15y = 10, can be rewritten as:

y = (4/3)x + (2/3)

This is a huge clue! It tells us that both lines have the same slope (4/3) and the same y-intercept (2/3). What does that mean, guys? It means we're not dealing with two intersecting lines here. We're dealing with the same line written in two different ways.

To see this more clearly with the elimination method, let's multiply the first equation (4x = 3y - 2) by 5. This gives us:

20x = 15y - 10

Now, let's rearrange this to match the form of the second equation:

-20x + 15y = 10

Wait a minute... that's exactly the second equation! If we try to add these equations together to eliminate a variable, we end up with 0 = 0. This is a true statement, but it doesn't give us specific values for x and y. It tells us something else very important, though.

Interpreting the Result

When we get an identity like 0 = 0, it means our system has infinitely many solutions. Why? Because both equations represent the same line. Any point that lies on this line will satisfy both equations. Think of it like this: you're trying to find where two lines intersect, but they're actually the same line! They overlap at every single point.

Infinite Solutions Explained

So, what does it really mean to have infinite solutions? It means that there isn't just one pair of values for 'x' and 'y' that makes both equations true. Instead, there's a whole range of values that work. Any 'x' and 'y' that satisfy the equation y = (4/3)x + (2/3) will be a solution to the system.

Visualizing Infinite Solutions

Imagine drawing the two lines on a graph. If you did, you'd only see one line because they're the same! Every single point on that line is a solution to both equations. This is different from a system with one solution, where the lines intersect at a single point, or a system with no solutions, where the lines are parallel and never intersect.

Conclusion

Alright, guys, we've cracked the case of this system of equations! We discovered that the equations 4x = 3y - 2 and -20x + 15y = 10 actually represent the same line. This means our system has infinitely many solutions. We used both rearrangement and the elimination method to arrive at this conclusion, highlighting the power of these techniques in understanding linear systems.

Remember, when you encounter a system of equations, always take a close look at what the equations are telling you. Sometimes, they might be hiding a little secret, like these did! Keep practicing, and you'll become a system-solving pro in no time!