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Straight-Line Equations: Slope-Intercept Form | Purplemath

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Covers the 'slope-intercept', or 'y = mx + b', form of straight-line equations.

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Practice telling whether or not the relationship between two quantities is proportional by reasoning about equivalent ratios.

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Number of solutions to equations (practice) | Khan Academy

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Practice telling whether an equation has one, zero, or infinite solutions. For example, how many solutions does the equation 8(3x+10)=28x-14-4x have?

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Solving algebra equations with variables on both sides is really tough. But, you will have no problems solving these equations after you follow the step-by-step instructions in these 3 examples.
So far, you've learned how to solve one-step equations, two-step equations, and equations with fractions. If you haven't mastered these skills or need a refresher, please go back and review these lessons before attempting this one! It will be so much easier if you have the background!

So, if you are ready, let's move on!

Solving algebra equations with variables on both sides look intimidating, but they really aren't that hard if you have the background skills. In fact, I think these types of equations are fun because it's like moving pieces of a puzzle around.

This is what you need to remember:

Solving Equations With Variables on Both Sides

If you have any fractions, get rid of those first by multiplying ALL terms by the denominator.

Use the distributive property if needed.

Your ultimate goal is to get all of the constants on one side of the equation and all of the variables on the other side of the equation. You can accomplish this by adding or subtracting terms on BOTH sides of the equation.

Let's take a look at a few examples.

Example 1 - Variables on Both Sides

Did you notice how we used all of the rules we learned in previous lessons?

Remember, our goal is to get all of the variables on one side and all the constants on the other side.

This next example shows how to solve if the distributive property is involved.

Example 2 - Working with the Distributive Property

Yes, this is a confusing topic, so if you are still having trouble, check out example 2 on video.

If you are still struggling with equations, head on over to our FREE Solving Equations E-course. You will find video tutorials for all lessons, practice problems, and step-by-step answer keys!

Now let's see what happens when we bring it all together and try to solve an equation that has fractions and variables on both sides!

Example 3 - Fractions and Variables on Both Sides

It doesn't look too hard, does it? Remember: it's like moving puzzle pieces around - all the variable terms on one side and all the constant terms on the other side!

Need Help With Your Homework?

Try this calculator for step by step answers (with subscription).

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Solving Algebra Equations with Variables on Both Sides

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Solving algebra equations with variables on both sides is really tough. But, you will have no problems solving these equations after you follow the step-by-step instructions in these 3 examples. So far, you've learned how to solve one-step equations, two-step equations, and equations with fractions.

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Algebra 1 / Exponents and exponential functions /

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2. Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p , where p is a positive rational number. Evaluate square roots of small...

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This is the video about how to evaluate square roots. The square root of a number is a number that you can square to get it, that is, a number that you can multiply by itself to get the number. So, 2 is a square root of 4, because 2 x 2 = 4, and 3 is a square root of 9, because 3 x 2 = 9. (-2) x 2 is also 4 and (-3) x 2 is also 9. Numbers that have square roots always have two, a positive one and a negative one, but the square root symbol means only the positive one, so we can have one answer to our problem. Negative numbers don't have square roots, because when you multiply numbers with like signs you get positive numbers.

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Quotient of Powers Rule

This is the formula for the Quotient of Powers Rule:

Now let's talk about what it means. The Quotient of Powers Rule assists with simplifying exponents. Let's first define some terms as they relate to exponents. When you have a number or variable raised to a power, the number (or variable) is called the base, while the superscript number is called the exponent or power.

In a division problem involving exponents, if the bases are the same, the problem can be simplified by subtracting the exponents. For example, take this problem: m^5 / m^3 = ?. Because the base ( m) is the same, you can simplify this problem by subtracting the exponents: 5 - 3 = 2. Therefore m^5 / m^3 = m^2.

Here is how this works.

Every m / m is equal to 1 and can be canceled out. For this problem, there are three of those 'sets,' which leave us with two ms in the numerator of the fraction. This simplifies to m^2.

Examples

You can use the Quotient of Powers Rule for simple or complex division problems. Let's do a few practice ones together. If you need more time to work them out, try pausing the video before you see the answers.

First, let's simplify f^7 / f^2.

Since the bases are the same, to solve this problem, just subtract the exponents. You should have gotten f^7 / f^2 = f^5 since 7 - 2 = 5.

You can only combine like terms. Those are the terms with the same bases. If it helps you to rearrange the problem so that terms with like bases are together, no problem. Just don't leave anything out. When you rearrange the terms, you get the following:

Then, just subtract the exponents for each term:

When you put it all together, the answer is ab^3 c.

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quotient of powers: property & examples - video & lesson transcript | study.com

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Quotient Of Powers: Property & Examples - Video & Lesson Transcript | Study.com

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This is the formula for the Quotient of Powers Rule: Now let's talk about what it means. The Quotient of Powers Rule assists with simplifying exponents. Let's first define some terms as they relate to exponents.

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Product of Powers Rule

The product of powers rule tells us that when you are multiplying two terms that have the same base, you can just add their exponents to find your answer.

Not sure what that means? Let's break it down.

Definition

The product of powers rule assists with simplifying exponents. Let's first define some terms as they relate to exponents. When you have a number or variable raised to a power, the number (or variable) is called the base, while the superscript number is called the exponent, or power.

In a multiplication problem involving exponents, if the bases are the same, the multiplication problem can be simplified by adding the exponents.

For example, look at this problem: m^3 * m^5 = ?

Because the base ( m) is the same, you can simplify the problem by adding the exponents: 3 + 5 = 8. Therefore, m^3 * m^5 = m^8.

Here is how this works:

Since the problem now asks us to multiply these two items together, we end up with the following: m^3 * m^5 = mmmmmmmm, which is equal to m^8 (or 8 m s multiplied together). Too bad they aren't M&Ms!

Examples

You can use the product of powers rule for simple or more complex multiplication problems. Let's apply it together.

Since the bases are the same, to solve this problem, we just add the exponents. Pause the video to try this before you see the answer onscreen.

Did you get f^2 * f^7 = f^9? Great! Let's do a more complicated problem.

Remember that you can only combine like terms. Those are the terms with the same bases. If it helps you to rearrange the problems so that terms with like bases are together, no problem. Just don't leave anything out. Since this is a multiplication problem, there is not as much to keep track of as there would be with addition or subtraction or even division.

Consider pausing the video to try this yourself before the answer is revealed for each step so you can check your progress.

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rule & property - video & lesson transcript | study.com

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Product Of Powers: Definition, Rule & Property - Video & Lesson Transcript | Study.com

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The product of powers rule tells us that when you are multiplying two terms that have the same base, you can just add their exponents to find your answer. Not sure what that means? Let's break it down. The product of powers rule assists with simplifying exponents.

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SparkNotes: Algebra I: Variation: Direct Variation

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A summary of Direct Variation in 's Algebra I: Variation. Learn exactly what happened in this chapter, scene, or section of Algebra I: Variation and what it means. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans.

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Here is a picture that really helped me with Direct Variation

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Here is a picture that really helped me with Direct Variation

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Comparing and ordering real numbers - Bing images

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Here is a direct variation video that really helped me because this definitely was NOT my strong suit in math

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Here is a direct variation video that really helped me because this definitely was NOT my strong suit in math

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Here is a website that helped me with comparing and ordering real numbers

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Here is a website that helped me with comparing and ordering real numbers

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Here is a picture that really helped me with comparing and ordering real numbers

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Here is a picture that really helped me with comparing and ordering real numbers

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Here is a video that really helped me on Comparing and Ordering Real numbers

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Here is a video that really helped me on Comparing and Ordering Real numbers

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comparing and ordering real numbers - Bing video

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Here is a very helpful website that gave me a better understanding of the real number system.

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Here is a very helpful website that gave me a better understanding of the real number system.

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Well, before this starts to sound like a Zen koan, let's look at how we use the numeral "0." Arab and Indian scholars were the first to use zero to develop the place-value number system that we use today. When we write a number, we use only the ten numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

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Here is a website that was very helpful to have a better understanding of problem solving stratigies

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Here is a website that was very helpful to have a better understanding of problem solving stratigies

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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication, or division? Then determine all the information that is being given to you in the question.

Mathematician George Pólya's book, " How to Solve It: A New Aspect of Mathematical Method," written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya's book and should help you untangle even the most complicated math problem.

Use Established Procedures

Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.

Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics.

If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.

Look for Clue Words

Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop.

If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.

Common clue words for addition problems:

Common clue words for subtraction problems:

Common clue words for multiplication problems:

Common clue words for division problems:

Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.

Read the Problem Carefully

This, of course, means looking for clue words as outlined in the previous section. Once you've identified your clue words, highlight or underline them. This will let you know what kind of problem you're dealing with. Then do the following:

Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?

What did you need to do in that instance?

What facts are you given about this problem?

What facts do you still need to find out about this problem?

Develop a Plan and Review Your Work

Based on what you discovered by reading the problem carefully and identifying similar problems you've encountered before, you can then:

Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.

If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.

If it seems like you've solved the problem, ask yourself the following:

Does your solution seem probable?

Does it answer the initial question?

Did you answer using the language in the question?

Did you answer using the same units?

If you feel confident that the answer is "yes" to all questions, consider your problem solved.

Tips and Hints

Some key questions to consider as you approach the problem may be:

1. What are the keywords in the problem?

2. Do I need a data visual, such as a diagram, list, table, chart, or graph?

3. Is there a formula or equation that I'll need? If so, which one?

Will I need to use a calculator? Is there a pattern I can use or follow?

Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.

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Tackle Just About Any Math Problem With These Strategies

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The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication, or division?

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Here is a picture on problem solving strategies that I found very helpful.

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Problem stratigies in math - Bing images

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