Line Equation And Gradient: Calculation And Drawing

by SLV Team 52 views

Hey guys! Today, we're diving into some cool math stuff about lines, gradients, and how to plot them. It might sound a bit technical, but trust me, it's super useful and kinda fun once you get the hang of it. We'll tackle finding gradients from equations and even plotting a line when you know its gradient and a point it passes through. Ready? Let's jump in!

1. Determine the Gradient of Each of the Following Line Equations

a. 3y - 2x = 5

Okay, so the first equation we've got is 3y - 2x = 5. To find the gradient, we need to get this equation into the form y = mx + c, where m is the gradient and c is the y-intercept. This form makes it super easy to spot the gradient. So, let's rearrange our equation.

First, we'll add 2x to both sides of the equation to isolate the term with y:

3y = 2x + 5

Next, to get y by itself, we'll divide every term by 3:

y = (2/3)x + 5/3

Boom! Now our equation is in the y = mx + c form. Looking at it, we can see that the coefficient of x is 2/3. That's our gradient! So, for the line 3y - 2x = 5, the gradient m is 2/3. This means that for every 3 units we move to the right on the graph, the line goes up by 2 units. Understanding gradient helps us visualize how steep the line is. A larger gradient means a steeper line, while a smaller gradient means a flatter line. Gradients can also be negative, which indicates that the line slopes downwards from left to right. This simple transformation allows us to quickly determine the slope of any linear equation, which is fundamental in many areas of mathematics and its applications.

b. 3x + 4y = 1

Alright, let's tackle the second equation: 3x + 4y = 1. Just like before, our mission is to get this equation into the y = mx + c form so we can easily identify the gradient. So, let's get rearranging!

First, we'll subtract 3x from both sides to isolate the term with y:

4y = -3x + 1

Now, to get y by itself, we'll divide every term by 4:

y = (-3/4)x + 1/4

There we go! The equation is now in the y = mx + c form. We can see that the coefficient of x is -3/4. That's our gradient for this line! So, for the line 3x + 4y = 1, the gradient m is -3/4. This tells us that for every 4 units we move to the right on the graph, the line goes down by 3 units. The negative sign indicates that the line slopes downwards from left to right. This is a crucial concept in understanding linear equations and their graphical representation. By converting the equation to slope-intercept form, we can easily identify the gradient and y-intercept, which are essential for graphing the line and analyzing its properties. The gradient, in particular, gives us valuable information about the steepness and direction of the line.

2. A Line with a Gradient of 3 Passes Through Point A(2, 4)

a. Draw the Line

Okay, so we know our line has a gradient of 3 and it passes through the point A(2, 4). Let's get this line plotted!

  • Plot the Point: First, plot the point A(2, 4) on your graph. This is our starting point.
  • Use the Gradient: The gradient is 3, which can also be written as 3/1. Remember, the gradient tells us how much the line goes up (or down) for every unit we move to the right. So, from point A, we move 1 unit to the right and 3 units up. This gives us our next point.
  • Find Another Point: Starting from A(2,4), move 1 unit to the right (to x=3) and 3 units up (to y=7). That gives us the point (3,7). Plot this point.
  • Draw the Line: Now, grab a ruler and draw a straight line through the points A(2, 4) and (3, 7). Extend the line in both directions as far as you need.

And there you have it! You've successfully plotted the line with a gradient of 3 that passes through the point A(2, 4). Visualizing a line this way helps us understand how the gradient affects its direction and steepness. Each step we take to the right increases the y-value by three, showing the direct relationship indicated by the slope. This graphical representation is a powerful tool for understanding linear equations and their applications in various fields, such as physics and engineering. The ability to plot such lines is essential for visualizing and interpreting mathematical relationships.

b. Determine the Second Coordinate of a Point on the Line if the First Coordinate is 5

Alright, so we know our line has a gradient of 3, passes through point A(2, 4), and now we want to find the y-coordinate when x is 5. We can use the point-slope form of a line equation to solve this. The point-slope form is:

y - y1 = m(x - x1)

Where:

  • (x1, y1) is a known point on the line (in our case, A(2, 4))
  • m is the gradient (which is 3)
  • (x, y) is any other point on the line

We want to find y when x = 5. So, let's plug in the values we know:

y - 4 = 3(5 - 2)

Now, let's simplify:

y - 4 = 3(3)

y - 4 = 9

To solve for y, we add 4 to both sides:

y = 9 + 4

y = 13

So, when x is 5, y is 13. That means the point (5, 13) lies on the line. This calculation shows how we can use the properties of a line—its gradient and a known point—to find any other point on that line. The point-slope form is a versatile tool for solving such problems, allowing us to determine unknown coordinates with ease. Understanding and applying this concept is vital for various mathematical and real-world applications, from predicting linear trends to designing physical structures. Thus, by using the gradient and a known point, we efficiently found the corresponding y-coordinate for a given x-coordinate on the line.

So, there you have it! We found gradients, plotted a line, and found a mystery coordinate. Keep practicing, and you'll become a line equation master in no time! ✨