Point (-4, 0) On Line Y = 9x - 2 A Verification
Hey guys! Today, we're diving into a fun little math problem that tests our understanding of linear equations and coordinate points. We're going to explore whether the point (-4, 0) satisfies the equation y = 9x - 2. It might sound a bit intimidating, but trust me, it's a straightforward process once you grasp the concept. We'll break it down step by step, so you can confidently tackle similar problems in the future.
Understanding the Basics: Points and Equations
Before we jump into the problem, let's quickly recap what points and equations represent in the world of mathematics. A point on a coordinate plane is defined by two values, an x-coordinate and a y-coordinate, written as (x, y). Think of it as an address on a map; it tells you the exact location of something. In our case, the point (-4, 0) means we move 4 units to the left along the x-axis and stay at the origin along the y-axis.
An equation, on the other hand, is a mathematical statement that shows the relationship between variables. Our equation, y = 9x - 2, describes a straight line on the coordinate plane. It tells us how the y-value changes as the x-value changes. The '9' is the slope, indicating the steepness of the line, and '-2' is the y-intercept, where the line crosses the y-axis.
So, what does it mean for a point to satisfy an equation? Simply put, it means that if we plug the x and y coordinates of the point into the equation, the equation holds true. The left side of the equation will equal the right side. If it doesn't, then the point doesn't lie on the line represented by the equation.
The Significance of Satisfying an Equation
Understanding whether a point satisfies an equation is crucial in various mathematical contexts. It's the foundation for:
- Graphing Lines: Knowing points that satisfy an equation helps us plot the line accurately on a graph. We can find several points that fit the equation and connect them to draw the line.
- Solving Systems of Equations: When dealing with multiple equations, finding the points that satisfy all equations simultaneously helps us solve for the unknowns.
- Real-World Applications: Linear equations are used to model numerous real-world scenarios, from calculating distances and speeds to predicting trends in data. Verifying points helps us ensure our models are accurate.
Testing the Point (-4, 0) Against the Equation y = 9x - 2
Alright, let's get our hands dirty and test whether the point (-4, 0) satisfies the equation y = 9x - 2. This is where the fun begins! We're going to substitute the x and y values of our point into the equation and see if it holds water.
Remember, our point is (-4, 0), which means x = -4 and y = 0. Now, let's plug these values into our equation:
y = 9x - 2 0 = 9*(-4) - 2
Now, let's simplify the right side of the equation:
0 = -36 - 2 0 = -38
Whoa there! We've hit a snag. The equation now reads 0 = -38, which is clearly not true. Zero and -38 are definitely not the same number. This tells us something important: the point (-4, 0) does not satisfy the equation y = 9x - 2.
Interpreting the Result: What Does It Mean?
So, we've established that the point (-4, 0) doesn't satisfy the equation. But what does this actually mean in a visual sense? Well, it means that the point (-4, 0) does not lie on the line represented by the equation y = 9x - 2.
Imagine graphing the line y = 9x - 2. It would be a steep line crossing the y-axis at -2. If you were to plot the point (-4, 0) on the same graph, you'd find that it sits somewhere off the line. It's like trying to fit a puzzle piece into the wrong spot; it just doesn't quite match up.
This understanding is fundamental in coordinate geometry. It helps us visualize the relationship between equations and their graphical representations. Points that satisfy an equation are the building blocks of the line or curve that the equation describes.
Why This Matters: Real-World Connections
You might be thinking,