Year 8 Plotting linear relationships
The position of any point in the plane can be represented by an ordered pair of numbers (x, y). These ordered pairs are called the coordinates of the point. To find the y-intercept, we substitute 0 for x in the equation, because we know that every point on the y-axis has an x-coordinate of 0. Once we do that, we can. single solution of a linear equation in two variables can in the Euclidean plane with a Cartesian coordinate system.
And it also goes through the origin. And it makes sense that it goes through an origin. Because in a proportional relationship, actually when you look over here, zero over zero, that's indeterminate form, and then that gets a little bit strange, but when you look at this right over here, well if X is zero and you multiply it by some constant, Y is going to need to be zero as well.
So for any proportional relationship, if you're including when X equals zero, then Y would need to be equal to zero as well. And so if you were to plot its graph, it would be a line that goes through the origin. And so this is a proportional relationship and its graph is represented by a line that goes through the origin. Now let's look at this one over here, this one in blue. So let's think about whether it is proportional.
And we could do the same test, by calculating the ratio between Y and X. So it's going to be, let's see, for this first one it's going to be three over one, which is just three.
Then it's gonna be five over two. Five over two, well five over two is not the same thing as three. So already we know that this is not proportional. We don't even have to look at this third point right over here, where if we took the ratio between Y and X, it's negative one over negative one, which would just be one. Let's see, let's graph this just for fun, to see what it looks like. When X is one, Y is three. When X is two, Y is five. X is two, Y is five. And when X is negative one, Y is negative one.
- Proportional relationships: graphs
When X is negative one, Y is negative one. And I forgot to put the hash mark right there, it was right around there. And so if we said, okay, let's just give the benefit of the doubt that maybe these are three points from a line, because it looks like I can actually connect them with a line.
Then the line would look something like this. The line would look something like this.
Linear equations in the coordinate plane
So notice, this is linear. This is a line right over here. But it does not go through the origin. So if you're just looking at a relationship visually, linear is good, but it needs to go through the origin as well for it to be proportional relationship. And you see that right here. This is a linear relationship, or at least these three pairs could be sampled from a linear relationship, but the graph does not go through the origin. And we see here, when we look at the ratio, that it was indeed not proportional.
So this is not proportional. Now let's look at this one over here. Let's look at what we have here. So I'll look at the ratios.
So for this first pair, one over one, then we have four over two, well we immediately see that we are not proportional. And then nine over three, it would be three. So clearly this is not a constant number here. We don't always have the same value here, and so this is also not proportional. But let's graph it just for fun.
Linear equation - Wikipedia
When X is one, Y is one. When X is two, Y is four. This actually looks like the graph of Y is equal to X squared. When you are dealing with data points plotted on a coordinate plane, a negative slope indicates a negative correlation and the steeper the slope, the stronger the negative correlation.
Consider working in your vegetable garden. If you have a flat of 18 pepper plants and you can plant 1 pepper plant per minute, the rate at which the flat empties out is fairly high, so the absolute value of m is a greater number and the line is steeper.
If you can only plant 1 pepper plant every 2 minutes, you still empty out the flat, but the rate at which you do so is lower, the absolute value of m is low, and the line is not as steep. Zero Slope When there is no change in y as x changes, the graph of the line is horizontal.
A horizontal line has a slope of zero. Undefined Slope When there is no change in x as y changes, the graph of the line is vertical. You could not compute the slope of this line, because you would need to divide by 0. These lines have undefined slope. Lines with the Same Slope Lines with the same slope are either the same line, or parallel lines. In all three of these lines, every 1-unit change in y is associated with a 1-unit change in x.
All three have a slope of 1. Solving Two-Step Linear Equations with Rational Numbers When a linear equation has two variables, as it usually does, it has an infinite number of solutions. Each solution is a pair of numbers x,y that make the equation true. Solving a linear equation usually means finding the value of y for a given value of x. To find ordered pairs of solutions for such an equation, choose a value for x, and compute to find the corresponding value for y.
Students may be asked to make tables of values for linear equations. These are simply T-tables with lists of values for x with the corresponding computed values for y. Two-step equations involve finding values for expressions that have more than one term. The terms in an expression are separated by addition or subtraction symbols.
To find a value for y given a value for x, substitute the value for x into the expression and compute. First, find the value of the term that contains x, then find the value of the entire expression.
This requires mirroring operations balancing on each side of the equation until y is by itself on the one side of the equation, set equal to an expression involving x. You can manipulate the equation in this way because of the equality properties: This equation is not in slope-intercept form. There are two ways to put it in slope-intercept form.