Solving Linear Relationships

[Mariela Laflam]

GRAPHINGLINEAR RELATIONS. The graph of a linear relation can be found by plotting at least two points. Two points that are especially useful for sketching the graph of a line are found with the intercepts. An x-intercept is an x-value at which a graph crosses the x-axis. A y-intercept is a y-value at which a graph crosses the y-axis. Since y = 0 on the x-axis, an x-intercept is found by setting y equal to 0 in the equation and solving for x. Similarly, a y-intercept is found by settingx=0 in the equation and solving for y.

Since y always equals -3, the value of y can never be 0. This means that the graph has no x-intercept. The only way a straight line can have no x-intercept is for it to be parallel to the x-axis, as shown in Figure 3.8. Notice that the domain of this linear relation is (-inf,inf) but the range is -3.

It can be shown, using theorems for similar triangles, that the slope slope is independent of the choice of points on the line. That is, the slope of a line is the same no matter which pair of distinct points on the line are used to find it.

Since the slope of a line is the ratio of vertical change to horizontal change, if we know the slope of a line and the coordinates of a point on the line, the graph of the line can be drawn. The next example illustrates this.

PARALLEL LINES Two distinct non vertical lines are parallel if and only if they have the same slope.

Slopes are also used to determine if two lines are perpendicular. Whenever two lines have slopes with a product of -1, the lines are perpendicular.

PERPENDICULAR LINES Two lines, neither of which is vertical, are perpendicular if and only if their slopes have a product of -1.

A straight line is often the best approximation of a set of data points that result from a real situation. If the equation is known, it can be used to predict the value of one variable, given a value of the other. For this reason, the equation is written as a linear relation in slope-intercept form. One way to find the equation of such a straight line is to use two typical data points and the point-slope form of the equation of a line.

Scientists have found that the number of chirps made by a cricket of a particular Species per minute is almost linearly related to the temperature. Suppose that for a particular species, at 68F a cricket chirps 124 times per minute, while at 80 F the cricket chirps 172 times per minute. Find the linear equation that relates the number of chirps to the temperature.

Traditionally, I have always taught evaluating expressions before teaching linear equations. But, I was recently given a remedial class of students that have to cover the bare minimums (and we have until mid-December to finish). Luckily, I have great flexibility with what I can do to the syllabus, so for the first time ever, I have completely cut out evaluating expressions since they wont even be tested on this on the final exam.

My question is more if anyone else has done this, or thinks this is not a good way to go. Most of my students in that particular class have ZERO to little formal math background, a lot of them did not even finish high school, and they barely get by with mean, median, mode, rounding, etc. I started equations with them today, and they seemed fine" for the most part. Of course, I also have spent the past week emphasizing positive and negative integer operations, so they are pretty OK with that so far. The textbook itself does not cover linear equations until after the section on evaluating expressions.

So my answer is: you do not need to specifically devote a day to "evaluating expressions", but you had better be sure that the students do achieve this outcome in tandem with solving equations. If you do not, then all of the work you have done will be meaningless.

Your context is students who have not learned much from the mathematics education they have been exposed to thus far. I guess, but can not be sure, that they have already been exposed to someone showing them algorithms and asking them to repeat. Because they are performing badly now, likely the teachers have tried to take this into account by teaching these things to them very slowly and thoroughly.

Given this, it is a tempting idea that you should do something, anything, else. What exactly this something else is seems to vary a bit, but the literature I have been exposed to recently has been all about open questions and questions with low threshold to doing something but with lots of room to expand.

A linear equation in two variables can be described as a linear relationship between x and y, that is, two variables in which the value of one of them (usually y) depends on the value of the other one (usually x). In this case, x is the independent variable, and y depends on it, so y is called the dependent variable.

Whether or not it's labeled x, the independent variable is usually plotted along the horizontal axis. Most linear equations are functions. In other words, for every value of x, there is only one corresponding value of y. When you assign a value to the independent variable, x, you can compute the value of the dependent variable, y. You can then plot the points named by each (x,y) pair on a coordinate grid.

Students should already know that any two points determine a line. So graphing a linear equation in fact only requires finding two pairs of values and drawing a line through the points they describe. All other points on the line will provide values for x and y that satisfy the equation.

The graphs of linear equations are always lines. However, it is important to remember that not every point on the line that the equation describes will necessarily be a solution to the problem that the equation describes. For example, the problem may not make sense for negative numbers (say, if the independent variable is time) or very large numbers (say, numbers over 100 if the dependent variable is grade in class).

In this equation, for any given steady rate, the relationship between distance and time will be linear. However, distance is usually expressed as a positive number, so most graphs of this relationship will only show points in the first quadrant. Notice that the direction of the line in the graph below is from bottom left to top right. Lines that tend in this direction have positive slope. A positive slope indicates that the values on both axes are increasing from left to right.

In this equation, since you won't ever have a negative amount of water in the bucket, the graph will show points only in the first quadrant. Notice that the direction of the line in this graph is top left to bottom right. Lines that tend in this direction have negative slope. A negative slope indicates that the values on the y-axis are decreasing as the values on the x-axis are increasing.

Again in this graph, we are relating values that only make sense if they are positive, so we show points only in the first quadrant. Moreover, in this case, since no polygon has fewer than 3 sides or angles and the number of sides or angles of a polygon must be a whole number, we show the graph starting at (3,3) and indicate with a dashed line that points between those plotted are not relevant to the problem.

The slope of a line tells two things: how steep the line is with respect to the y-axis and whether the line slopes up or down when you look at it from left to right. More technically speaking, slope tells you the rate at which the dependent variable is changing with respect to the change in the independent variable.

The equation of a line can be written in a form that makes the slope obvious and allows you to draw the line without any computation. If students are comfortable with solving a simple two-step linear equation, they can write linear equations in slope-intercept form. The slope-intercept form of a linear equation is y = mx + b. In the equation, x and y are the variables. The numbers m and b give the slope of the line (m) and the value of y when x is 0 (b). The value of y when x is 0 is called the y-intercept because (0,y) is the point at which the line crosses the y-axis.

When a line slopes up from left to right, it has a positive slope. This means that a positive change in y is associated with a positive change in x. The steeper the slope, the greater the rate of change in y in relation to the change in x. A slope of 6 is steeper than a slope of 1, which is in turn steeper than a slope of 1/6. When the line represents real-world data points plotted on a coordinate plane, a positive slope indicates a positive correlation, and the steeper the slope, the stronger the positive correlation.

Consider a linear equation where the independent variable g is gallons of gas used and the dependent variable d is the distance traveled in miles. If you drive a big, old car, you get poor gas mileage. The amount of miles traveled is low relative to the amount of gas consumed, so the value m is a low number. The slope of the line is fairly gradual. If instead you drive a light, efficient car, you get better gas mileage. You travel more miles relative to the same amount of gas consumed, so the value of m is greater 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.

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 the line represents real-world 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 a line that represents the number of peppers left to plant after minutes spent gardening. If the garden can fit 18 pepper plants, and you plant 1 pepper plant per minute, the rate at which the garden flat empties out is fairly high, so the absolute value of m is a greater number, and the line is steeper. If instead you only plant 1 pepper plant every 2 minutes, you still empty out the garden flat, but the rate at which you do so is lower. The absolute value of m is lower (1/2 instead of 1), and the line is not as steep.

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