Category Archives: math

Factor Differences of Square VIDEO

Factoring a polynomial using a difference of squares

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Solving a System of Non-Linear Equations

Solving a system of non-linear equations means that at least one of the equations is not linear. For example, if one equation has an exponent it may be a parabola or a circle. With this no shape that is not linear it involves slightly different expectations.

Solving a system of non-linear equations is similar to solving a system with linear equations with one difference. The difference is that with nonlinear equations you can have more than one solution. What this means is that the lines that are the equations can intersect in more than one place. However, it is also possible they do not intersect. How many solutions depends on the lines involved.

For example, if one equation is a circle/parabola and the other is a line there can be 0-2 solutions. If one equation makes a circle and the other makes a parabola there can be up to 4 solutions. Two circles or two parabolas can make a multitude of solutions

The steps for solving a nonlinear system are the same. Therefore, in this post, we will demonstrate how to solve a system of non-linear equations using the substitution and elimination methods.

Substitution

The substitution method is when we plug one equation into the variable of the other equation. Below is our system of equations.

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The first thing to notice is that the top equation would make a circle if you graphed it. That is why this is a non-linear system. To solve we take the second equation and substitute it for y. Below we find the values for x

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Now we complete the system by finding the values for y.

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THerefore are ordered pairs are (0, -3) and (1,0). These are the two points at which the equations intersect if you were to graph them.

Elimination

Elimination involves making the coefficients of one of the variables opposite so that when they are added together they cancel each other out. By removing one variable you can easily solve for the other. Below is the system of equation we want to solve.

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The top equation makes a circle while the bottom one makes a parabola. This means that we can have as many as four solutions for this system. To solve this system we will multiply the bottom equation by -1. This will allow us to remove the x variable and then solve for y. Below are the steps.

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Now we simply solve for y.

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Now we can take these values for y to solve for x.

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The order pairs are as follows

  • (-2,0)
  • (2,0)
  •   (√3, -1)
  • (-√3, -1)

Conclusion

From this, you can see that non-linear equations can be solved using the same approaches. Understanding this is key to many other fields of math such as data science and machine learning.

Logarithmic Models

A logarithm is the inverse of exponentiation. Depending on the situation one form is better than the other. This post will explore logarithms in greater detail.

Converting Between Exponential and Logarithmic Form

There are times when it is necessary to convert an expression from exponential to logarithmic and vice versa. Below is an example of who the expression is rearranged form logarithm to exponential.

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The simplest way to explain I think is as follows

  • for the logarithm, the exponent (y) and the base (a) are on opposite sides of the equal sign
  • For the exponent form, the exponent (y) and base (a) are on the same side of the equal sign.

Here is an example using actual numbers1.png

As you can see the exponent 3 and the base 2 are on opposite sides of the equal sign for the logarithmic form but er together for the exponential form.

When the base is e (Euler’s Number) it is known as a natural logarithmic function. e is the base rate growth of a continual process. The application of this is limitless. When the base is ten it is called a common logarithmic function.

Logarithmic Model Example

Below is an example of the application of logarithmic models

Exposure to noise above 120 dB can cause immediate pain and damage long-term exposure can lead to hearing loss. What aris the decimal level of a tv with an intensity of 10^1 watts per square inch. 

First, we need the equation for calculating the decibel level.

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Now we plug in the information into the word problem for I and solve

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Our tv is dangerously loud and should include a warning message.  We dropped the negative sign because you cannot have negative decibel level.

Conclusion

Logarithms are another way to express exponential information and vice versa. It is the situation that determines which to use and the process of concert an expression from one to another is rather simple. In terms of solving actual problems, it is a matter of plugging numbers into an equation and allowing the calculator to work that allows you to find the answer.