Why Study Math? Linear Equations and the Point-Slope Form

As we saw in the article "Why Study Math? - Linear Equations and Slope-Intercept Form," linear equations or functions are some of the more basic ones studied in algebra and basic mathematics. Here we are going to take a look at and examine another common way of writing linear equations: the point-slope form.

As the name implies, the point-slope form for the equation of a line depends on two things: the slope, and a given point on the line. Once we know these two things, we can write the equation of the line. In mathematical terms, the point-slope form of the equation of the line which passes through the given point (x1, y1) with a slope of m, is y - y1 = m(x - x1). (The 1 after the x and y is actually a subscript which allows us to distinguish x1 from x and y1 from y.)

To show how this form is used, take a look at the following example: Suppose we have a line which has slope 3 and passes through the point (1, 2). We can easily graph this line by locating the point (1, 2) and then use the slope of 3 to go 3 units up and then 1 unit to the right. To write the equation of the line, we use a clever little device. We introduce the variables x and y as a point (x,y). In the point-slope form y - y1 = m(x - x1), we have (1, 2) as the point (x1, y1). We then write y - 2 = 3(x - 1). By using the distributive property on the right hand side of the equation, we can write y - 2 = 3x - 3. By bringing the -2 over to the right side, we can write
y = 3x -1. If you have not already recognized it, this latter equation is in slope-intercept form.

To see how this form of the equation of a line is used in a real world application, take the following example, the information of which was taken from an article that appeared in a newspaper. It turns out that temperature affects running speed. In fact, the best temperature for running is below 60 degrees Fahrenheit. If a person ran optimally at 17.6 feet per second, he or she would slow by about 0.3 feet per second for every 5 degree increase in temperature above 60 degrees. We can use this information to write the linear model for this situation and then calculate, let us say, the optimal running pace at 80 degrees.

Let T represent the temperature in degrees Fahrenheit. Let P represent the optimal pace in feet per second. From the information in the article, we know that the optimal running pace at 60 degrees is 17.6 feet per second. Thus one point is (60, 17.6). Let's use the other information to determine the slope of the line for this model. The slope m is equal to the change in pace over the change in temperature, or m = change in P/change in T. We are told that the pace slows by 0.3 feet per second for every increase in 5 degrees above 60. A decrease is represented by a negative. Using this information we can calculate the slope at -0.3/5 or -0.06.

Now that we have a point and the slope, we can write the model which represents this situation. We have P - P1 = m(T - T1) or P - 17.6 = -0.06(T - 60). Using the distributive property we can put this equation into slope-intercept form. We obtain P = -0.06T + 21.2. To find the optimal pace at 80 degrees, we need only substitute 80 for T in the given model to get 16.4.

Situations like these show that math is really used to solve problems that occur in the world. Whether we are talking about optimal running pace or maximal profits, math is the key to unlocking our potential toward understanding the world around us. And when we understand, we are empowered. What a nice way to exist!

Why Study Math? Linear Systems and the Substitution Method - Part I

After having read my previous articles on lines and their representative equations, we are now going to look at these objects two at a time, in which case we have what is known as a linear system. When we have two lines, and we try to graph them on the same coordinate plane, three things can happen: one, the lines may never intersect, in which case they are parallel; two, the lines may coincide, in which case they are identical; or three, the lines can intersect in one point, the unique solution of the system. Here we are going to look at how to find that unique solution using the method known as substitution.

In my article, "Why Study Math? - Linear Equations and Slope-Intercept Form," we talked about the standard form for the equation of the line, that is Ax + By = C. When we have a linear system, or pair of lines, whose solution we are looking to find, we usually write them in standard form. One of the advantages of standard form is that it allows us to solve the system more easily when we try to eliminate one of the variables. This method, known as elimination or linear combinations, is another method that we will look at in an upcoming article. For now, we are going to focus on the method of substitution, which
works particularly nicely when one of the equations is in slope-intercept form.

The method of substitution, as the name implies, allows us to substitute one of the variables by its value in terms of the other, and then solve for the variable. For example, take the following system: y = 2x + 1 and 3x - 2y = 5. The first equation tells us what y is in terms of the variable x. We use this in the second equation and replace y with 2x + 1. Doing this we obtain,
3x - 2(2x + 1) = 5. Now that we have an equation in x only, we can solve to get x = -7. We then plug this value in the first equation to get y = 2(-7) + 1 or -13. The unique solution of this system then is -7, -13, which is a point and the unique point where, if we were to graph these two lines on the same coordinate plane, the two lines intersect.

Although this method usually gives students trouble, there really is no difficulty with it if you keep your head on straight and your pencil sharpened. For example, take the following system:
3x - 5y = 4, and y = -3x + 10. By using the second equation in the first, we obtain
3x - 5(-3x + 10) = 4. Simplifying, we have 3x + 15x -50 = 4. Rearranging terms, we get
18x = 54 or x = 3. We use this value of x to get y = -3(3) + 10 or y = 1.

There really is no difficulty with this method if you keep your figures neat and precise. Just keep your eraser handy for an error or two and remember that errors are okay because the more mistakes you make, the better you are going to get. Remember: if at first you don't succeed, just do another problem. Happy problem solving.

Why study mathematics? linear equations and the form of point-slope

As we saw in the article "Why Study Mathematics - linear equations and slope-shape, linear equations or functions, only some of the most fundamental areas of mathematics and basic algebra studied. Here we will look to consider and another common way to write linear equations: the point-slope form.

As the name suggests, the shape point-slope equation of a line depends on two things: the slope and a point on the line. Once we know the two things we canWrite the equation of the line. y1 = m - Mathematically, the shape of the point-slope equation of the line passing through the point (x1, y1) with a slope of m, y (x - x1). (The 1 in x and y is actually an index that distinguishes us from X1 and Y1 of Y allows X)

To show how this form is used, take a look at the following example: Suppose you have a line, the track 3, and the point (1, 2). We can easily graph this line byThe point (1, 2) and then use the 3-3 slope unit and then go right one unit. To write the equation of the line, using a smart device. There are variables x and y as a point (x, y). In point-slope form y - y1 = m (x - x1), we have (1, 2) as the point (x1, y1). We then write y - 2 = 3 (x - 1). Using the distributive property on the right side of the equation, we can write y - 2 = 3x - 3 -2 By right, we can write
y = 3x -1. If you have not already guessed, is to make this last equation in slope-intercept.

To see how this form of equation of a line in a real world "is used, take the following example, the information taken from an article in a newspaper. It turns out that the temperature influences the speed. In fact, the optimum temperature for operation below 60 degrees Fahrenheit. If a person was at his best, at 17.6 meters per second, he or sheslow down to about 0.3 meters per second for every 5 degree increase in temperature above 60 degrees. We can use this information to write the linear model of this situation, and then calculate, for example, the optimal speed of 80 degrees.

Let T is the temperature in degrees Fahrenheit. Let P represent the optimal speed in meters per second. The information in this article we know that the optimal speed of 60 degrees 17.6 meters per second. This point(60, 17.6). We use other information to determine the slope of the line for this model. The slope m is equal to the speed variation of the temperature change or M = P / T change We are told that the pace slowed from 0.3 meters per second for every increase of 5 degrees above the 60 ° A decrease is represented by a negative. Using this information we can calculate the slope to -0.3 / 5 or -0.06.

Now that we have a point and the slope, we can write the modelrepresents this situation. We have P - P 1 = m (T - T1) or P - 17.6 = -0.06 (T - 60). Using the distributive property can use this equation in slope-intercept. We get P =- 0.06T + 21.2. To find the optimal speed of 80 degrees, we need to get a compensation of only 80 for the Model T on 16.4.

Situations like these show that math is really used to solve problems that arise in the world. If we're talking about the optimum speed and maximum profit is the key to the mathematicalUnlock our potential for understanding the world around us. And if we understand, we are the power. There was a nice way!

Why study mathematics? - Solution of linear by linear combinations

Now that we have seen how to solve a system of linear equations using the substitution method, you move to a more convenient method known as linear combinations. With this method, in addition, subtraction reputation, it eliminates a variable from a most appropriate one of equations. We can eliminate a variable and solve for others. Once done, you use the other equation to solve for the other variables.

This method can be done algorithmically and thenThese are the steps for solving a system of linear combinations:

Step 1: Arrange the equations with like terms in columns.

Step 2: Multiply one or both of the coefficients of the equation in order to dilute the variables are opposites of one of the.

Step 3: Add the equations of the previous step. Combine like terms eliminated one of the variables and fixes to others.

Step 4: Replace the value of the resultantto solve the equations in a previous step and the other variable.

Step 5: Check the solution in each of the original equations.

To illustrate the algorithm, we solve the following system: 4x + 3y = 16 and 2x - 3y = 8 Initially, these two equations is in columns, so that variables such as line-up. So we

4x + 3y = 16

2x - 3y = 8

As you can see that the coefficients of the y-contrast conditions, there is no need to multiply the equationsGet this form. And so with only two, disappear so that the terms-y. We 6x = 24 Solving for x we have x = 4 Substituting this value in the first equation, for example, gives four (4) + 3y = 16 or 16 + 3y = 16 3y = 0 or y = 0. Check with the inclusion of these values generated in each original equation a true statement. Thus, the solution is x = 4 y = 0 or the point (4, 0) as the intersection of these two lines in a coordinate system.

Let us see howthe method of linear combinations of a model of historical problem. According to legend, the famous greek mathematician Archimedes, the ratio between the weight of an object and its volume is used to determine that there is fraud in the manufacture of a golden crown. The way this was done by applying the principle of displacement volume. You see, when a crown of pure gold, so the same volume the same amount to replace the gold. The number that follows, the concept ofThe density is also used. By definition, the density of an object equals its mass divided by volume. Gold has a density of 19 grams per cubic centimeter. Silver has a density of 10.5 grams per cubic centimeter. We are made so that the problem follows to use.

Problem: Suppose a crown of gold and silver with some suspicion, he weighed 714 grams and had a volume of 46 cubic centimeters. What percentage of the crown was silver?

To resolve this problem, we observe thatthe volume of the volume of gold silver, more must be equal if the total volume of 46. Since they are the densities of gold and silver, white, and we know that the density of times the corresponding density, we have that golden moment of the density, the volume of gold, more silver is the silver density times band the total weight. We let G = Gold S = volume of silver and volume. Now we can translate the problem in mathematics and a linear system.

We have a G + S + G = 46 and 19 = 10.5S 714th Puttingcolumns in these equations, we

G + S = 46

19G + = 10.5S 714th

Now multiply the first equation of -19 to opposite coefficients for G. So we get

-19G +-19S = -874

19G + = 10.5S 714th

Adding the two equations we have - 8.5S = -160. Dividing both sides by -8.5, we have S = 18.8, we set up in 19. The volume of silver is 19 cubic centimeters, and the percentage of silver in the crown is 19/46 or 41% for the nearest percentage.Remember, this method next time someone tries to pawn on you from pure gold, when in fact the reality is quite different. Watch out for Fool's Gold 's!

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Why Study Math? Linear Systems and the Substitution Method - Part II

As we are quickly learning from my series of articles on lines and their applications, the power of these mathematical objects should not be taken for granted because of their simplicity. Lines and more specifically, linear systems, find important applications in the fields of telecommunications, signal processing, and automatic control, the last field of which deals with such interesting things as the programming, guidance, and control of ballistic missiles. In the first article in this series, we examined how to solve a linear system by the method of substitution. Here we will look at some basic problems which employ such linear systems.

Let us look at the following example which deals with museum admissions at, let us say, the Museum of Natural History in New York City. Suppose that in one day, this museum collected $1590 from 321 people admitted to see its splendors. The price of each adult admission is $6. People between the ages of 4-17 pay the child admission of $4. Let us calculate how many of each type, adult and child, were admitted to the museum.

Linear systems give us a neat way of solving this problem. To appreciate the power of linear systems and the method of substitution, which we are going to use to solve this problem,
try to hazard a guess as to how you would figure this out. You will quickly see that there is no convenient way to get the number of adults and the number of children that were admitted. Yet by creating some linear models in the form of a system, we can quickly arrive at the answer to this problem.

Let us start with a verbal model and then translate this into mathematics. This is a convenient and helpful strategy which will allow us to solve the problem more readily. We have from the information that the number of adults plus the number of children is equal to 321. We also have that the number of adults times the price of an adult admission plus the number of children times the price of a child's admission is equal to the total amount collected.

If we let x represent the number of adults and y represent the number of children, we can translate the verbal model into a linear system of equations. Since the price for adults is $6 and the price for children $4 and the total number of people attending 321, we have
x + y = 321 and 6x + 4y = 1590. Notice that both of these equations are in standard form. We can easily take the first equation and put into slope-intercept form by writing
y = -x + 321 (moving x to the other side). We now substitute into the second equation to get
6x + 4(-x + 321) = 1590. Simplifying, we have 2x = 306, or x = 153. Using this value of x to get y, we have y = 168.

Thus 153 adults and 168 children attended the museum. And since all the children were so good in doing something educational with their parents, they all went home with very nice souvenirs. Now isn't that a nice way to spend the day!

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