Finding Points On A Line The Same Y-Intercept As 7x + 3y = 9

Hey everyone! Today, we're diving into a fun math problem that involves finding points on a line. This isn't just any line, though. It's a line that passes through a specific point and shares the same y-intercept as another line. Sounds intriguing, right? Let's break it down step by step so we can tackle this problem like pros. So, the main goal here is to identify which table displays four points residing on a line. This line must confidently pass through the coordinates (1, 1). Here's the catch – it should also flaunt the same y-intercept as the line represented by the equation 7x + 3y = 9. To ace this task, we'll need to roll up our sleeves and find the y-intercept of the given equation. Then, we'll craft the equation for our new line. After that, it's all about checking which points from the table fit the equation like a glove.

Understanding the Basics

Before we jump into the solution, let's quickly recap some fundamental concepts. Remember, a line can be defined by its slope and y-intercept, or by two points on the line. The slope tells us how steep the line is, and the y-intercept is the point where the line crosses the y-axis. The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. This form is super useful because it directly gives us the slope and y-intercept of the line. Now, when we talk about the y-intercept, we are basically referring to the point where the line intersects the y-axis. This happens when x is 0. So, if we know the y-intercept, we know one point on the line: (0, b), where b is the y-coordinate of the intercept. The y-intercept is a crucial characteristic of a line because it, along with the slope, completely defines the line's position and orientation on the coordinate plane. Knowing the y-intercept helps us visualize and graph the line accurately.

Finding the Y-Intercept

The first step in solving our problem is to find the y-intercept of the given line, which is defined by the equation 7x + 3y = 9. To do this, we need to rewrite the equation in slope-intercept form (y = mx + b). This involves isolating y on one side of the equation. Let's walk through the process:

1. Start with the equation: 7x + 3y = 9.
2. Subtract 7x from both sides: 3y = -7x + 9.
3. Divide both sides by 3: y = (-7/3)x + 3.

Now we have the equation in slope-intercept form. We can see that the slope (m) is -7/3 and the y-intercept (b) is 3. This means the line crosses the y-axis at the point (0, 3). So, the y-intercept is 3. This is a key piece of information because our new line must have the same y-intercept. Finding the y-intercept is a fundamental skill in algebra and is essential for understanding and manipulating linear equations. By converting the equation to slope-intercept form, we can easily identify the y-intercept and use it to graph the line or solve related problems. In our case, knowing the y-intercept allows us to define one of the characteristics of the line we are trying to find, which is a crucial step in solving the problem. So, with the y-intercept in hand, we can move on to the next step of figuring out the equation of our new line.

Determining the New Line's Equation

Alright, guys, we've nailed down the y-intercept, which is 3. Now, we need to figure out the equation of the line that passes through the point (1, 1) and has a y-intercept of 3. Remember, we're aiming for the slope-intercept form: y = mx + b. We already know b (the y-intercept) is 3, so our equation looks like y = mx + 3. To fully define the line, we need to find the slope (m). We can use the point-slope form of a linear equation to find the slope. The point-slope form is given by: y - y1 = m(x - x1), where (x1, y1) is a point on the line. We know the line passes through (1, 1), so we can plug in x1 = 1 and y1 = 1. We also know that the line passes through the y-intercept (0, 3). We can use these two points to calculate the slope (m) using the formula: m = (y2 - y1) / (x2 - x1). Let's use the points (1, 1) and (0, 3):

• m = (3 - 1) / (0 - 1)
• m = 2 / -1
• m = -2

So, the slope of our line is -2. Now we can plug this into our equation y = mx + 3, giving us y = -2x + 3. This is the equation of the line that passes through (1, 1) and has the same y-intercept as the line 7x + 3y = 9. With this equation, we can now test points to see if they lie on the line. Finding the equation of a line given certain conditions is a fundamental skill in algebra and geometry. By using the point-slope form and the slope-intercept form, we can determine the equation of a line that meets specific criteria, such as passing through a given point and having a particular y-intercept. This skill is crucial for solving a variety of problems in mathematics and other fields.

Testing the Points

Okay, we've got our equation: y = -2x + 3. Now comes the fun part – testing the points from the table to see which ones lie on this line. To do this, we'll plug in the x-coordinate of each point into our equation and see if we get the corresponding y-coordinate. If the equation holds true, then the point lies on the line. Let's say we have a table with the following points:

• (-1, 5)
• (2, -1)
• (3, -3)
• (4, -5)

We'll test each point one by one. For the point (-1, 5), plug in x = -1 into our equation:

• y = -2(-1) + 3
• y = 2 + 3
• y = 5

Since we got y = 5, the point (-1, 5) lies on the line. Let's try the next point, (2, -1):

• y = -2(2) + 3
• y = -4 + 3
• y = -1

Again, the equation holds true, so (2, -1) is also on the line. For the point (3, -3):

• y = -2(3) + 3
• y = -6 + 3
• y = -3

This point checks out as well. Finally, let's test (4, -5):

• y = -2(4) + 3
• y = -8 + 3
• y = -5

This point also lies on the line. So, in this example, all four points lie on the line y = -2x + 3. The table that shows these points is the one we're looking for. Testing points against an equation is a fundamental technique in algebra and geometry. It allows us to verify whether a point satisfies the equation of a line or curve, which is essential for graphing and solving problems involving geometric figures. By plugging in the coordinates of a point into the equation, we can determine if the point lies on the line, which is a crucial step in many mathematical applications.

Conclusion

So, there you have it! We've successfully navigated through the process of finding points on a line with a specific y-intercept. We started by understanding the basics of linear equations and slope-intercept form. Then, we found the y-intercept of the given line and used that information to determine the equation of our new line. Finally, we tested the points to see which ones lie on the line. This problem is a great example of how different concepts in algebra and geometry come together to solve a single problem. By mastering these concepts, you'll be well-equipped to tackle similar challenges in the future. Keep practicing, and you'll become a math whiz in no time! Remember, math isn't just about numbers and equations; it's about problem-solving and critical thinking. So, embrace the challenge, and enjoy the journey!