Using Linear Models – Graphing a Linear Model, Interpret the y-intercept

This is the second post in the series on working with linear models. Here is the link to the first post: Using Linear Models – Graphing a Linear Model Using the x- and y-Intercepts

In this post I will discuss writing a linear model or linear equation to represent a problem situation. Once the equation is written, I will identify and interpret the y-intercept. The embedded video will model all of the steps necessary to solve the problem. Here is the problem.

A spring has a length of 8 cm when a 20-g mass is hanging at the bottom end. Each additional gram stretches the spring another 0.15 cm. Write an equation for the length y of the spring as a function of the mass x of the attached weight. Graph the equation. Interpret the y-intercept.

Understanding the first sentence of the problem and how the values are related is key to writing the equation. From the first sentence it can be concluded that the length of the spring will depend on the amount of weight attached to the spring. This means the weight is the independent variable and the length of the spring is the dependent variable. The problem even goes as far as stating that length should be represented by the variable y and the weight is represented with the variable x. x traditionally is the independent variable and y is the dependent variable, as is the case in this problem.

The second sentence reads, “Each additional gram stretches the spring another 0.15 cm.” This relationship between the additional weight and the amount of stretch is a rate of change. The spring will expand 0.15 cm for every 1 gram added. Remember, the slope of a line is also a rate comparing the change in y-values over the change in x-values and because the y-values are representing the length of the spring and the x-values are representing the weight added, we can ascertain that the slope is 0.15cm/1gram or m = 0.15.

It is given in the problem, the spring has a length of 8 cm when a 20-g mass is hanging at the bottom end. This can be written as the ordered pair (20, 8 ) and can be used with the slope m = 0.15 to write a linear equation for this problem.  To write this equation, it will be easiest to start with the point-slope form of a linear equation:

y – y1 = m(x – x1)  –>  Point-Slope Form of a Linear Equation (1)

Substituting the point (20,8) and the slope m = 0.15 into equation 1,

y – 8 = 0.15(x – 20).         (2)

To you can make a graph from this form, but since interpreting the y-intercept is a part of the problem using some algebra to put the equation into slope-intercept form will be useful.  Remember, the slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept. To put equation 2 into slope-intercept form, first 0.15 must be distributed to give:

y – 8 = 0.15x – 3.               (3)

Finally, adding 8 to both sides of equation 3 will give the slope-intercept form:

Y = 0.15x + 5.      (4)

The y-intercept is (0,5) and means that when the spring has now weight attached , it is 5 cm long. As before, see the video for the graphing portion of this problem.

A related example to this spring problem, what mass would be needed to stretch the spring to a length of 9.5 cm? This is very simple to complete if you understand the meaning of the variables. Y represents the length of the spring and x represents the amount of weight attached to the spring. For this problem, you must find HOW MUCH WEIGHT must be attached to stretch the spring 9.5 cm. Thus, you are given the length of the spring or the y-value and you must solve for the x-value.

To complete this task, substitute 9.5 for y in equation 4 to get:

9.5 = 0.15x + 5.           (5)

To solve for x, subtract 5 from both sides,

4.5 = 0.15x           (6)

And divide both sides by 0.15 to find,

x = 30        (7)

Relating this answer to the problem, it will take 30 grams to stretch the spring to 9.5 cm.

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Using Linear Models – Graphing a Linear Model Using the x- and y- Intercepts

This video is about using linear models. The exercise models how to write a linear model or linear equation for a given problem situation.

Suppose an airplane descends at a rate of 300 ft/min from an elevation of 8000 ft. Write and graph an equation to model the plane’s elevation as a function of the time it has been descending. Interpret the intercept at which the graph intersects the vertical axis.

To solve this problem, it is essential that one understands independent and dependent variables. Remember, when graphing on the coordinate plane, the independent variable is graphed along the x-axis and the dependent variable is graphed along the y-axis. As with most problems involving time, TIME is the independent variable and the countdown will start when the plane starts its decent. The plane’s elevation is the dependent variable. It has a distinct starting value of 8000 feet and it will be changing at a rate of -300 feet per minute. To find the plane’s elevation at any given time:

x = time elapsed measured in minutes

y = plane’s elevation measured in feet

The elevation equals the rate of descent times the time plus the starting elevation. The previous sentence translates into equation 1.

y = -300x + 8000                (1)

To graph this linear model, it would be best to use the x- and y-intercepts. The y-intercept will be          (0, 8000). This represents the beginning of the decent. The elapsed time is zero and the plane is at its starting elevation of 8000 feet. The x-intercept will is found by substituting 0 in for y in equation 1, giving equation 2.

0 = -300x + 8000                (2)

To find the x-intercept or the amount of time it will take the plane to land, add 300x to both sides of (2) to get:

300x = 8000         (3).

Divide both sides (3) by 3000:

x = 80/3 = 26.7 seconds            (4).

There are now two ordered pairs that can be graphed: (0, 8000) and (26.7, 0). See the video for the actual graphing.

Geometry – Properties of Parallel Lines 3.1 Part 2

This lesson investigates and use the alternate interior angles theorem, the alternate exterior angles theorem, the corresponding angles postulate, the same side interior angles theorem and the same side exterior angles theorems. The other post titled, Geometry – Properties of Parallel Lines, would not allow me to put up my other video, so here is the 1st video lesson on properties of parallel lines.

Parallel and Perpendicular Lines – Geometry – 3.3

This lesson is about the parallel lines theorem, the perpendicular to same line theorem and the perpendicular to one line theorem. Now, these are shorter names that I have given these theorems.

The Parallel Lines Theorem

If two lines are parallel to the same line,
then they are parallel to each other.

The Perpendicular to Same Line Theorem

In a plane, if two lines are perpendicular to the same line,
then they are parallel to each other.

The Perpendicular to One Line Theorem

In a plane, if a line is perpendicular to one of two parallel
lines, the it is also perpendicular to the other.

The first problem covered in the video is working with frame making. It is not the best I have ever done, but I will be putting something together in the near future to add to this
post.

The second problem is a paragraph proof involves proving two lines parallel. The diagram involves three lines that appear to be parallel that are in the order from top to
bottom, lines a, b and c. There are two transversals, but only one is labeled line s. Line is is perpendicular to lines a and c and lines a and b are parallel. We must prove c is parallel to b. From given, it can be established that b is perpendicular to s because of the Perpendicular to One Line Thm. Now we can state that c is parallel to b because of the
Perpendicular to Same Line Thm. QED

If you have a specific question, please ask. Cite your book, I might have it and I can show the specific problem. Also, give your best description of the problem that you can. You must quote the question from your book, which means you have to give the name and author with copyright date. I have used digital images of problems I have worked out by hand for the Algebra 2 portion of my blog.

Proving Lines Parallel – Geometry – 3.2

Proving Lines Parallel – Geometry

This lesson investigates and use the converse of alternate interior angles theorem, the converse of alternate exterior angles theorem, the converse of corresponding angles postulate, the converse of same side interior angles theorem and the converse of same side exterior angles theorem.

The first problem in the video covers determining which pair of lines would be parallel with the given information. You are given that two same-side exterior angles are supplementary. There two pairs of lines that appear to parallel. You must determine which pair is parallel with the given information. One could argue that both pairs are parallel, because it could be used, but the problem is ONLY asking for what can be proved with the given information.

Much like the lesson on Properties of Parallel Lines the second problem models how to find the value of x that allow two lines to be parallel. You much write an equation
based on how the angles are related. The problem in the video show how to solve a problem that involves converse of alternate interior angles theorem, converse of alternate exterior angles theorem, converse of corresponding angles postulate. Which means an equal relationship. Remember, the supplementary relationship, where the sum of the given angles is 180 degrees.

Similar to the first problem, the third problem has you determining which lines are parallel, but the diagram is of a wooden frame with a diagonal brace. Two alternate interior angles are marked congruent. Remember, you are only asked for which sides are parallel by the given information.

If you have a specific question, please ask. Cite your book, I might have it and I can show the specific problem. Also, give your best description of the problem that you can. You must quote the question from your book, which means you have to give the name and author with copyright date. I have used digital images of problems I have worked out by hand for the Algebra 2 portion of my blog.

Geometry – Properties of Parallel Lines – 3.1

Geometry – Properties of Parallel Lines

This lesson investigates and use the alternate interior angles theorem, the alternate exterior angles theorem, the corresponding angles postulate, the same side interior angles theorem and the same side exterior angles theorems.

The use of a two column proof to show that 2 angles are supplementary, but the angles are not able to be proved with a theorem or postulate. The proof takes three or four steps.

Another problem in the video lesson below is finding the measure of angles when give two parallel lines and a transversal. There are two types of these problems. The first type of finding the measure of an angle is when you are given the actual measure of the angle such as 50 degrees. You are able to find the measure of every angle formed by 2 parallel lines and a transversal, when you know the measure of one. If one angle measures 50 degrees, all of the every angle in the diagram will be 50 or 130 degrees. All angles are either congruent or supplementary in this situation.

The other type of find the measure of an angle problem is when you are given two or more algebraic expressions instead of real numbers as the measurements. For example you are given the

measure of angle 1 = 3x + 7  (1)

measure of angle 6 = 5x + 15 (2).

You need to find the value of x that makes the lines parallel. It will be

3x + 7 = 5x + 15, (3)

Equation 3 is the result of the angles having a congruent relationship as with alternate interior angles, alternate exterior angles and corresponding angles. If the angles are same side interior angles or same side exterior angles, then the equation would be:

(3x + 7) + (5x + 15) = 180 (4)

If you have a specific question, please ask. Cite your book, I might have it and I can show the specific problem. Also, give your best description of the problem that you can. You must quote the question from your book, which means you have to give the name and author with copyright date. I have used digital images of problems I have worked out by hand for the Algebra 2 portion of my blog.

Here is the second video.

Negative and Zero Exponents – Pre Algebra

Negative and Zero Exponents – Pre Algebra

This is a video math lesson discussing negative and zero exponents. The examples in this lesson model how to simplify expressions with negative and zero exponents. Lesson 4.6.

This is is a two part video math lesson discussing negative and zero exponents. Negative and zero exponents are defined in a general manner. The examples in this lesson model how to simplify expressions with negative and zero exponents. Lesson 4.6.

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