lesson 3 homework practice solve equations with rational coefficients

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Lesson 3 Homework Practice Solve Equations with Rational Coefficients

You’re now diving into Lesson 3 , where your mission will be to tackle equations with rational coefficients. The term ‘rational coefficients’ is intimidating. But don’t worry! We’re here to break it down. A rational coefficient is a fancy term for a number that can be formulated as a ratio of two integers. It provides a logical framework to tackle complex equations.

Overview of solving equations with rational coefficients

First up, let’s take a clear look at equations. You can think about equations like balanced see-saws. Whatever you do on one side has to be balanced out on the other side. Equations with rational coefficients can be easily simplified by manipulating the fractional coefficients to isolate the variable.

Step 1:  Identify the rational coefficient. The fraction in the equation is multiplied by the variable (e.g., 2/3x = 4).

Step 2:  To isolate your variable, you will perform the inverse operation to eliminate the rational coefficient. You’ll multiply both sides of your equation by the reciprocal of your rational coefficient.

Example:  Let’s use the equation above. If we multiply both sides of 2/3x = 4 by 3/2 (the reciprocal of 2/3), we get 1x = 6. And voila, you have your solution.

In just a few steps, you’ve tackled equations with rational coefficients. Don’t let challenges discourage you in mathematics. It involves estimation, trial and error, and practice. But you can master it. Keep practicing, be persistent, and continue to hone your problem-solving skills . Remember, every mathematician learns at their own pace. You got this!

Practice is the key here, so nail down these steps and tackle additional problems independently. Happy solving!

The Concept of Rational Coefficients

You’ve made it to Lesson 3 , where you’ll tackle problems including rational coefficients. It might seem daunting at first, but with a systematic approach and a little practice, you’ll quickly learn how to navigate this mathematical territory.

Explanation of rational coefficients and their role in equations

In algebra, a rational coefficient is a ratio of two integers with a zero denominator. It is important to remember that these coefficients, just like any number in an algebraic equation, can be manipulated through the usual multiplication, division, addition, and subtraction.

Now, you ask, how do rational coefficients affect the equation? These coefficients play a pivotal role in determining the solution to the equation.

Here is how you can approach it:

Step 1:  Simplify the equation. If you have fractions in the equation, sometimes called rational numbers , making them whole numbers is easier. You can multiply every term with the least common denominator (LCD).

Step 2:  The next step is to use simple math techniques to find the variable. It is fundamentally the same as solving any basic algebraic equation –addition, subtraction, multiplication, and division are your friends.

Step 3:  Remember to check if your solution is correct. Substitute the solution into the original equation and see if both sides balance.

Remember:  The key to mastering equations with rational coefficients is understanding and accurately manipulating these coefficients to find your solution.

Remember, the key to success is consistent practice. Each equation you tackle improves your skill and confidence. Solving equations with rational coefficients will be easy. Good luck!

Let’s recap:

Techniques for Solving Equations with Rational Coefficients

As an aspiring mathematician or someone trying to polish their math skills , navigating the world of equations with rational coefficients can sometimes seem daunting. Worry not! Let’s simplify the process into two easy-to-follow steps: clearing fractions and cross-multiplication.

Step-by-step methods for solving equations with rational coefficients

Step 1:  Identify the equation with a rational coefficient. Rational coefficients are fractions expressed as a ratio of two integers.

Step 2:  Apply one of the strategies below to solve the equation.

• Method 1: Clearing Fractions

Wherever you see fractions, simplify them!

Step 1:  Multiply each term in the equation by the fraction’s denominator to clear the fractions.

Step 2:  Carry out the arithmetic operations and solve the equation.

Step 3:  Check your work by substituting the solution into the original equation. If it makes the equation true, then your solution is correct.

• Method 2: Cross-Multiplication

When your solution is a proportion, this method works well.

Step 1:  Use cross-multiplication if two fractions are set equal.

Step 2:  Multiply the terms diagonally and set the products equal.

Step 3:  Solve for the variable.

Step 4:  Verify your solution by plugging it into the original equation.

Please make no mistake: the key to mastering mathematical equations is through practice, and lots of it. Please don’t avoid making mistakes; use them to strengthen your understanding and sharpen your skills. Get your homework and apply the strategies you learned. You can solve these problems if you keep practicing. With unwavering dedication and practice, you can conquer any equation that comes your way.

Examples and Practice Problems

You’ll come across equations with rational terms as you learn math. At first, it might seem challenging, but after reading this blog, you can do it like a pro!

Solving equations with rational coefficients through examples and practice problems

As you embark on your mathematical journey, you are bound to encounter equations with rational coefficients. By the end of this guide, you will be a pro at solving these types of equations.

• Example 1: Clearing fractions and solving for the variable

Take this equation:  3/4x = 6. First, clear out the fraction by multiplying both sides of the equation by 4. You’ll get 3x = 24. Now, try solving for ‘x’ by dividing 24 by 3. Your answer will be x = 8.

Now let’s practice:

• Try solving 2/7y = 6.
• Keep in mind the steps followed in the example.
• Clear the fraction, isolate the variable, and solve for ‘y’.
• Example 2: Cross-multiplication to solve for the variable

Here’s our equation:  4/5 = x/10. Cross-multiplication involves multiplying the fraction’s numerator on the left by the fraction’s denominator on the right and vice versa. This gives us 4 10 = 5 x, which simplifies to 40 = 5x.

Proceed with the operation:  Solve for ‘x’ by dividing both sides by 5, which leaves us with x = 8.

Time to practice:  Try the same steps with the equation 3/4 = y/12. Follow this example above, and you can solve this equation effectively.

In both types of equations, remember always to check your answers. Substituting your calculated value into the original equation should give you a true statement. These tutorials can guide you in solving complex equations and hone your skills.

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• Lesson 3: Solving Rational Equations

Hi Everyone!

On this page you will find some material about Lesson 3. Read through the material below, watch the videos, and follow up with your instructor if you have questions.

Table of Contents

In this section you will find some important information about the specific resources related to this lesson:

• the learning outcomes,
• the section in the textbook,
• the WeBWorK homework sets,
• a link to the pdf of the lesson notes,
• a link to a video lesson.

Learning Outcomes.

• Solve a rational equation.
• Check the potential solutions.

Topic . This lesson covers Section 5.5: Solving Rational Equations.

WeBWorK . There is one WeBWorK assignment on today’s material:

FractionalEquations

Lesson Notes.

Video Lesson.

Video Lesson 3 (based on Lesson 3 Notes)

Warmup Questions

These are questions on fundamental concepts that you need to know before you can embark on this lesson. Don’t skip them! Take your time to do them, and check your answer by clicking on the “Show Answer” tab.

Warmup Question 1

Solve $$\dfrac{3x+5}{x}=2,$$ and check your answer.

Show Answer 1

$$\dfrac{3x+5}{x}=2$$

Multiplying both sides by $x$ gives

$$3x+5 = 2x$$

Check : $$\dfrac{3(-5)+5}{-5}=\dfrac{-10}{-5} =2$$

Warmup Question 2

What is the LCD (least common denominator) of

$$\dfrac{1}{x^4}, \dfrac{x^3}{x^2-1}, \quad\text{and}\quad\dfrac{x-1}{2x(x+1)}?$$

Show Answer 2

The LCD is $2x^4(x^2-1)$.

If you are not comfortable with the Warmup Questions, don’t give up! Click on the indicated lesson for a quick catchup. A brief review will help you boost your confidence to start the new lesson, and that’s perfectly fine.

Need a review? Check Lesson 1 .

Quick Intro

This is like a mini-lesson with an overview of the main objects of study. It will often contain a list of key words, definitions and properties – all that is new in this lesson. We will use this opportunity to make connections with other concepts. It can be also used as a review of the lesson.

A Quick Intro to Solving Rational Equations

Key Words. Solution to an equation, rational equation, LCD (least common denominator), checking potential solutions.

The first step in solving a rational equation is to clear the denominators. For example, in the equation

$$\dfrac{2}{x}=\dfrac{x}{2}$$

the LCD is $2x$. Multiplying both sides by $2x$ clears the denominators.

\begin{align*} \dfrac{2}{x}\cdot 2x&=\dfrac{x}{2}\cdot 2x\\2\cdot 2& = x\cdot x\\4 &=x^2\\x^2&=4\\x&=\pm 2\end{align*}

The potential solutions are $x=\pm 2$. We now need to check them.

$\bullet$ Check: $x=-2$

\begin{align*}\dfrac{2}{-2} &\stackrel{?}{=}\dfrac{-2}{2} \\-1 &= -1 \quad\checkmark \end{align*}

$\bullet$ Check: $x=2$

\begin{align*}\dfrac{2}{2} &\stackrel{?}{=}\dfrac{2}{2}\\1&=1 \quad\checkmark \end{align*}

The solution set is $\{-2,2\}$.

Now watch the video lesson to see that, depending on the equation, another approach can be taken. You will see that it is important to always check the solutions.

Video Lesson

Many times the mini-lesson will not be enough for you to start working on the problems. You need to see someone explaining the material to you. In the video you will find a variety of examples, solved step-by-step – starting from a simple one to a more complex one. Feel free to play them as many times as you need. Pause, rewind, replay, stop… follow your pace!

A description of the video

In the video you will see how to solve

$$\dfrac{x}{x-1}-\dfrac{1}{x} = \dfrac{3}{2}$$

in two ways.

Try Questions

Now that you have read the material and watched the video, it is your turn to put in practice what you have learned. We encourage you to try the Try Questions on your own. When you are done, click on the “Show answer” tab to see if you got the correct answer.

Try Question 1

$$\dfrac{12}{x}-\dfrac{12}{x-5}=\dfrac{2}{x}.$$

$$\dfrac{12}{x}-\dfrac{12}{x-5}=\dfrac{2}{x} $$

The LCD is $x(x-5)$. We multiply both sides of the equation by $x(x-5)$.

\begin{align*}\dfrac{12x(x-5)}{x}-\dfrac{12x(x-5)}{x-5} &=\dfrac{2x(x-5)}{x} \\12(x-5) – 12x&= 2(x-5) \\12x-60-12x& = 2x-10 \\-60& =2x-10 \\-50 &= 2x\\2x &= -50\\x&=-25\end{align*}

The potential solution is $x=-25$. The next step is to check it.

$\bullet$ Check: $x=-25$

\begin{align*}\dfrac{12}{x}-\dfrac{12}{x-5}&=\dfrac{2}{x} \\\dfrac{12}{-25}-\dfrac{12}{-25-5} & \stackrel{?}{=}\dfrac{2}{-25} \\ -\dfrac{12}{25} + \dfrac{12}{30} &\stackrel{?}{=} – \dfrac{2}{25} \\-\dfrac{12\cdot 6}{150} + \dfrac{12\cdot 5}{150} &\stackrel{?}{=} – \dfrac{2}{25} \\ \dfrac{-72+60}{150}  &\stackrel{?}{=} – \dfrac{2}{25}\\\dfrac{-12}{150}  &\stackrel{?}{=} – \dfrac{2}{25} \\-\dfrac{2}{25} &\stackrel{?}{=} – \dfrac{2}{25} \quad\checkmark \end{align*}

Therefore the solution set is $\{-25\}$.

You should now be ready to start working on the WeBWorK problems. Doing the homework is an essential part of learning. It will help you practice the lesson and reinforce your knowledge.

It is time to do the homework on WeBWork:

When you are done, come back to this page for the Exit Questions.

Exit Questions

After doing the WeBWorK problems, come back to this page. The Exit Questions include vocabulary checking and conceptual questions. Knowing the vocabulary accurately is important for us to communicate. You will also find one last problem. All these questions will give you an idea as to whether or not you have mastered the material. Remember: the “Show Answer” tab is there for you to check your work!

• What is the difference between an expression and an equation? 
• What does it mean to solve an equation? 
• Is it necessary to check your answer if you know you have not made a mistake?  Explain.

$\bigstar$ Solve $$\quad\dfrac{y}{y+3}+\dfrac{3}{y-3} = \dfrac{18}{y^2-9}.$$

Show Answer

The LCD is $y^2-9$ or $(y-3)(y+3)$.

\begin{align*}\dfrac{y}{y+3}+\dfrac{3}{y-3} & = \dfrac{18}{y^2-9}\\(y-3)(y+3)\left(\dfrac{y}{y+3}+\dfrac{3}{y-3}\right) &= (y-3)(y+3)\dfrac{18}{y^2-9}\\\dfrac{y(y-3)(y+3)}{y+3}+\dfrac{3(y-3)(y+3)}{y-3}& =\dfrac{18(y-3)(y+3)}{y^2-9} \\y(y-3)+3(y+3) & = 18 \\y^2-3y+3y+9 & = 18\\y^2 &= 9\\y& = \pm\sqrt 9\\ y & = \pm 3\end{align*}

The potential solutions are $y=\pm 3$. We will now check them.

$\bullet$ Check: $y=3$

$$\dfrac{3}{3+3}+\dfrac{3}{3-3} \stackrel{?}{=} \dfrac{18}{3^2-9}$$

Two denominators are zero for the value of $y=3$.

$\bullet$ Check: $y=-3$

\[\dfrac{-3}{-3+3}+\dfrac{3}{-3-3} \stackrel{?}{=} \dfrac{18}{(-3)^2-9}\]

Two denominators are zero for the value of $y=-3$.

The solution set is the empty set, $\{\}$.

Need more help?

Don’t wait too long to do the following.

• Watch the additional video resources.
• Talk to your instructor.
• Form a study group.
• Visit a tutor. For more information, check the tutoring page .

Lessons Menu

• Lesson 1: Properties of Integer Exponents & Addition and Subtraction of Rational Expressions
• Lesson 2: Complex Fractions
• Lesson 4: Roots and Rational Exponents
• Lesson 5: Simplifying Radical Expressions & Addition and Subtraction of Radicals
• Lesson 6: Multiplication of Radicals
• Lesson 7: Division of Radicals and Rationalization
• Lesson 8: Solving Radical Equations
• Lesson 9: Complex Numbers
• Lesson 10: Solving Equations by Using the Zero Product Rule
• Lesson 11: Square Root Property and Completing the Square & Quadratic Formula
• Lesson 12: Applications of Quadratic Equations
• Lesson 13: Graphs of Quadratic Functions & Vertex of a Parabola
• Lesson 14: Distance Formula, Midpoint Formula, and Circles & Perpendicular Bisector
• Lesson 1: Systems of Linear Equations With Three Variables
• Lesson 16: Nonlinear Systems of Equations in Two Variables
• Lesson 17: Angle Measure and Special Triangles & The Trigonometry of Right Triangles
• Lesson 18: Solving Right Triangles & Applications of Static Trigonometry
• Lesson 19: Angle Measure in Radian & Trigonometry and the Coordinate Plane
• Lesson 20: Unit Circles
• Lesson 21: Graphs of Sine and Cosine
• Lesson 22: Fundamental Identities and Families of Identities
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• Lesson 24: Oblique Triangles and The Law of Sines & The Law of Cosines
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