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SAT Subject Test Mathematics Level 2: Tutoring Solution24 chapters | 231 lessons

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will go over equivalent equations and how to use the rules of equivalent equations to write an equation in standard form. This process is extremely useful when you are working as a team member on a specific problem.

Imagine this: you're working with two other students, Andrea and Susan, on a problem in math class. You are trying to find an equation to represent a word problem. You all work separately to solve the problem and then compare answers. You came up with the equation 2*x* - *y* = -1. Andrea came up with the equation *y* = 2*x* + 1. Lastly, Susan came up with *y* - 3 = 2(*x* - 1) as her equation. You all think that your answer is correct, so why did you all come up with different answers? You go to ask the teacher, and the teacher tells you that, in fact, you are all correct! What? How are you all correct if you got different answers?

The answer to this conundrum lies in equivalent equations. **Equivalent equations** are equations that have the same solution set, but may look different. When it comes to equivalent equations, you can manipulate one equation to make it look exactly the same as the other equation. For example, consider yours and Andrea's solutions. Andrea got *y* = 2*x* + 1. Try subtracting 2*x* from both sides of that equation, and then multiplying both sides by -1, which you can see playing out below:

You see that you can manipulate Andrea's equation to become your equation. You can do the same with any two of the equations that you three came up with because they are all equivalent equations. The following rules of equivalent equations show how we can manipulate an equation to produce an equivalent equation in the form we would like.

There are many different ways to write an equation. We are going to concentrate on just one way of writing an equation, and that's in standard form. Starting with linear equations, let's look at how to write an equation in standard form.

**Linear equations** are equations that, when graphed, form a line. The highest exponent of any variable in a linear equation is 1. You can see some of the examples below:

Notice that the equations that you, Andrea, and Susan got are linear equations. The standard form of a linear equation is as follows:

In the standard form of a linear equation, the *A* should be positive, so if you get it into standard form and the *A* is negative, multiply both sides of the equation by a negative to make it positive.

This tells us that if we want to put a linear equation in standard form, we must manipulate the equation so that the left-hand side of the equation has the *x* and *y* terms, with the *x* term coming first, and the right-hand side of the equation has a constant.

Look back at the three equations you and your friends came up with in our example. Which one is in standard form?

If you said your equation was the one in standard form, you are correct! Notice that when we manipulated Andrea's equation to put it in the form of your equation, we were actually putting Andrea's equation in standard form using our equivalent equation rules.

A **polynomial equation** is an equation in which one or both of the sides of the equation forms a polynomial. Some examples of polynomial equations are as the ones below:

When it comes to putting polynomial equations in standard form, we want to get zero on the right-hand side of the equation, and all non-zero terms in descending order (meaning the exponents go from highest to lowest) on the left-hand side of the equation.

Once again, if we want to get a polynomial equation in standard form, we simply manipulate it using the equivalent equation rules to get it in this form. For example, suppose you have the following polynomial equation:

Let's use the rules of equivalent equations to get it in standard form.

That wasn't so hard, was it?

Let's take a couple moments to review some of the important information that we've learned about writing an equation in standard form. **Equivalent equations** are equations that have the same solution set, but may look different. We can manipulate equations to form equivalent equations in a desired form using the equivalent equation rules.

There is a standard form of **linear equations** - which are equations that, when graphed, form a line, and **polynomial equations** - which are equations in which one or both of the sides of the equation forms a polynomial. Linear equations can be described quite simply: it's *Ax* + *By* = *C*, where *A*, *B*, and *C* are constants. With polynomial equations, we want to get 0 on the right-hand side of the equation and all non-0 terms in descending order on the left-hand side of the equation.

We can use the equivalent equation rules to manipulate an equation and put it in standard form. This is really useful when you're working with other people on a problem. It eases communication and verifies that you are all on the same page. Now that you're familiar with standard form, the next time your teacher gives you a problem to work on with other students, you can all agree on putting it in standard form so that when you compare answers, you know if you all got the same answer.

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SAT Subject Test Mathematics Level 2: Tutoring Solution24 chapters | 231 lessons

- What are Polynomials, Binomials, and Quadratics? 4:39
- How to Add, Subtract and Multiply Polynomials 6:53
- Multiplying Binomials Using FOIL and the Area Method 7:26
- Multiplying Binomials Using FOIL & the Area Method: Practice Problems 5:46
- How to Factor Quadratic Equations: FOIL in Reverse 8:50
- Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient 7:35
- How to Divide Polynomials with Long Division 8:05
- How to Use Synthetic Division to Divide Polynomials 6:51
- Dividing Polynomials with Long and Synthetic Division: Practice Problems 10:11
- How to Solve a Quadratic Equation by Factoring 7:53
- How to Solve Quadratics That Are Not in Standard Form 6:14
- How to Complete the Square 8:43
- Completing the Square Practice Problems 7:31
- How to Use the Quadratic Formula to Solve a Quadratic Equation 9:20
- Using the Quadratic Formula to Solve Equations with Literal Coefficients 5:37
- Solving Problems using the Quadratic Formula 8:32
- How to Solve Standard Form: Rules & Practice
- How to Find Slope in Standard Form 6:01
- How to Write a Number in Standard Form 6:45
- How to Write an Equation in Standard Form 5:56
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