Modelling Linear Relationships
Answer to The graph below shows a linear relationship between x and y. Note that the axes are to scale. Determine the constant rat. What's the Difference Between Equations and Functions? Which graph shows the relationship between x and y? Graphing a Basic (Linear) Function. Introduction to modelling linear relationships with tables, graphs and using a By trial and error, we look for a relationship between the values of x and y. We can it represents a straight line relationship between the coordinates of the points.
Or at least for these three points that we've sampled, and we'll say, well, maybe it's always the case, for this relationship between X and Y, or if you wanted to write it another way, you could write that Y is equal to one half X.
Now let's graph this thing. Well, when X is one, Y is one half. When X is four, Y is two. When X is negative two, Y is negative one. I didn't put the marker for negative one, it would be right about there. And so if we say these three points are sampled on the entire relationship, and the entire relationship is Y is equal to one half X, well the line that represents, or the set of all points that would represent the possible X-Y pairs, it would be a line.
It would be a line that goes through the origin. Because look, if X is zero, one half times zero is going to be equal to Y. And so let's think about some of the key characteristics. One, it is a line. This is a line here. It is a linear relationship. 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.
Proportional relationships: graphs
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.
MathSteps: Grade 7: Linear Equations: What Is It?
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. 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. 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. If you drive a big, heavy, old car, you get poor gas mileage. The rate of change in miles traveled is low in relation to the change in gas consumed, so the value m is a low number and the slope of the line is fairly gradual.
If you drive a light, efficient car, you get better gas mileage. The rate of change in the number of miles you travel is higher in relation to the change in gas consumed, so the value of m is a greater number and the line is steeper.
Both rates are positive, because you still travel a positive number of miles for every gallon of gas you consume. Negative Slope When a line slopes down from left to right, it has a negative slope. This means that a negative change in y is associated with a positive change in x. 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.