Finding The Best Approximate Solution: Linear Equations

Hey math enthusiasts! Today, we're diving into the world of linear equations and finding the best approximate solution for a system of these equations. Specifically, we're focusing on the equations $y = 1.5x - 1$ and $y = 1$. Our mission? To figure out which of the given points – $(0.33, 1)$, $(1.33, 1)$, $(1.83, 1)$, and $(2.33, 1)$ – is the closest match to the actual solution. Let's break it down, shall we?

Understanding Linear Equations and Solutions

Alright, before we jump into the nitty-gritty, let's refresh our memory on what linear equations and their solutions are all about. In simplest terms, a linear equation is an equation that, when graphed, forms a straight line. Think of it like this: you've got an equation like $y = mx + b$, where 'm' is the slope, and 'b' is the y-intercept. The solution to a system of linear equations is the point (or points) where the lines represented by those equations intersect. This point satisfies all equations in the system simultaneously. Basically, it's the spot where all the lines "agree" with each other. In our case, we have two equations: $y = 1.5x - 1$ and $y = 1$. The solution to this system is the point $(x, y)$ that makes both equations true.

Now, the equations we're dealing with are pretty straightforward. The second equation, $y = 1$, tells us that the y-coordinate of our solution is 1. This means the solution point has to sit somewhere on the horizontal line where $y = 1$. The first equation, $y = 1.5x - 1$, is slightly more complex; it shows a line with a slope of 1.5 and a y-intercept of -1. The point where these two lines intersect is the exact solution to the system. However, since we are looking for the best approximate solution, we'll check which of the provided points gets us closest to that actual intersection point.

To find the true solution, we can use a couple of methods. One is substitution: since we know $y = 1$, we can plug that value into the first equation. So, $1 = 1.5x - 1$. Solving for 'x', we add 1 to both sides to get $2 = 1.5x$, and then divide by 1.5 to find that $x = 1.33$ (approximately). Therefore, the exact solution is approximately $(1.33, 1)$. We can also solve this graphically by plotting both equations on a coordinate plane and finding their intersection. So, we've established how to find the exact solution, but what if we can't do that? That's where the approximate solutions come into play.

Evaluating the Proposed Solutions

Okay, let's put on our detective hats and investigate each of the proposed solutions to see which one fits the bill best. We have four potential candidates: $(0.33, 1)$, $(1.33, 1)$, $(1.83, 1)$, and $(2.33, 1)$. Our goal is to determine which one is closest to the actual solution, which we've already found to be approximately $(1.33, 1)$.

For each point, we'll substitute the x-value into the first equation ($y = 1.5x - 1$) and see how close the calculated y-value is to the actual y-value of the point (which is 1). The point with the smallest difference, or the smallest error, between the calculated and the actual y-value, will be our best approximate solution. Let's get to it!

• Point 1: (0.33, 1) Substitute x = 0.33 into $y = 1.5x - 1$: $y = 1.5(0.33) - 1$ $y = 0.495 - 1$ $y = -0.505$

  The calculated y-value is -0.505, which is quite far from the actual y-value of 1. The error is $|1 - (-0.505)| = 1.505$.

• Point 2: (1.33, 1) Substitute x = 1.33 into $y = 1.5x - 1$: $y = 1.5(1.33) - 1$ $y = 1.995 - 1$ $y = 0.995$

  The calculated y-value is 0.995, which is very close to the actual y-value of 1. The error is $|1 - 0.995| = 0.005$. This is looking promising!

• Point 3: (1.83, 1) Substitute x = 1.83 into $y = 1.5x - 1$: $y = 1.5(1.83) - 1$ $y = 2.745 - 1$ $y = 1.745$

  The calculated y-value is 1.745, which is further away from the actual y-value of 1. The error is $|1 - 1.745| = 0.745$.

• Point 4: (2.33, 1) Substitute x = 2.33 into $y = 1.5x - 1$: $y = 1.5(2.33) - 1$ $y = 3.495 - 1$ $y = 2.495$

  The calculated y-value is 2.495, which is quite a distance from the actual y-value of 1. The error is $|1 - 2.495| = 1.495$.

Determining the Best Approximate Solution

Alright, we've crunched the numbers and now it's time to crown the winner! Based on our calculations, the point (1.33, 1) is the clear frontrunner. When we plugged x = 1.33 into the equation $y = 1.5x - 1$, we got a y-value of approximately 0.995, which is incredibly close to the actual y-value of 1. The error, or the difference between the expected and calculated y-values, was only 0.005. This is significantly smaller compared to the errors from the other points, which means that $(1.33, 1)$ is the most accurate approximate solution. We can clearly see that the other points, (0.33, 1), (1.83, 1), and (2.33, 1) have larger errors and are therefore less accurate approximations of the solution to the system of linear equations. Therefore, the answer is definitely $(1.33, 1)$.

Remember that in this context, the approximation comes from the fact that we are testing pre-defined points to check which fits best. If the task was to actually solve the equations, then we could use different methods like substitution, elimination, or graphing.

So, there you have it! We've successfully navigated the world of linear equations, calculated, evaluated, and ultimately found the best approximate solution. Hope you guys enjoyed this math adventure! Keep practicing, and you'll become math wizards in no time. Until next time, happy solving!