Solving Systems of Equations With Fractions and Decimals

The Organic Chemistry Tutor
17 Jan 201805:41
EducationalLearning
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TLDRThe video script provides a detailed walkthrough for solving systems of linear equations involving fractions and decimals. The first method demonstrated is clearing fractions by multiplying through by a common denominator, followed by using the elimination method to find the values of x and y. The second scenario introduces substitution as the preferred method when one variable is isolated. The script also covers eliminating decimals by multiplying through by a power of ten to simplify calculations. The process is clearly explained with step-by-step instructions, culminating in the solution of the equations.

Takeaways
  • 🧩 When solving equations with fractions, it's helpful to eliminate them by multiplying through by the denominator.
  • 📚 To clear fractions in the first equation, multiply every term by 3, simplifying the equation to a form without fractions.
  • 🔍 In the second equation, multiplying every term by 4 eliminates the fractions, making it easier to work with.
  • 📈 The elimination method is recommended when variables align, which can be achieved by multiplying the second equation by 3 to match the Y coefficients.
  • 🌟 After aligning the equations, adding them allows for the cancellation of Y variables, simplifying the process to solve for X.
  • 📝 Once the value of X is found, it can be substituted back into one of the original equations to solve for Y.
  • 🔑 If one variable is isolated in one of the equations, substitution is the preferred method for solving the system.
  • 🔢 When dealing with decimals, multiplying every term by a power of 10 can help eliminate them and simplify the equation.
  • 📉 Distributing and combining like terms is crucial for simplifying the equation after substitution.
  • 📌 Solving for X first simplifies the process of finding Y by substituting the value of X into the other equation.
  • 🎯 The final solution to the system of equations is given as a pair of values (X, Y), in this case, (8, 2).
Q & A

  • What is the first step to simplify an equation with fractions?

    -The first step is to clear away all fractions by multiplying every term in the equation by the denominator of the fraction to convert it into a whole number.

  • How do you eliminate fractions in the equation 1/3x + 3y = 18?

    -To eliminate fractions, multiply every term in the equation by 3, which is the denominator of the fraction 1/3, resulting in x + 3y = 18.

  • What is the purpose of multiplying the second equation by 4 to clear away fractions?

    -Multiplying the second equation by 4 helps to clear away the fractions 1/2 and 1/4, simplifying the equation to 2x - y = 8.

  • Why is the elimination method preferred when x and y are aligned?

    -The elimination method is preferred because it simplifies the process of solving the system of equations by adding or subtracting the equations to cancel out one variable.

  • How do you use the elimination method to solve for x in the given equations?

    -Multiply the second equation by 3 to align the y terms, then add the two equations to cancel out the y variable, allowing you to solve for x.

  • What is the value of x after using the elimination method on the equations provided?

    -After adding the equations, the value of x is found to be 6 by dividing 42 by 7.

  • How do you find the value of y once you have the value of x?

    -Substitute the value of x into one of the original equations and solve for y, which in this case is found to be 4.

  • What method should be used when one of the equations has y isolated?

    -When y is isolated in one of the equations, the substitution method is preferred as it simplifies the process of finding the value of x.

  • How do you handle decimals when using the substitution method?

    -To handle decimals, multiply every term in the equation by a power of 10 that matches the number of decimal places, in this case, 10, to eliminate the decimals.

  • What is the final solution for the system of equations involving decimals?

    -After substituting and eliminating decimals, the solution is found to be x = 8 and y = 2.

Outlines
00:00
📚 Solving Equations with Fractions and Decimals

This paragraph discusses methods for solving equations involving fractions and decimals. It begins by suggesting clearing fractions from the first equation by multiplying through by a number that will eliminate the denominators, such as multiplying by 3/1 to simplify 1/3x and 3 times y to 3y and 18. For the second equation, multiplying by 4 is recommended to clear fractions, resulting in 2x, -1y, and 8. The elimination method is then introduced as a strategy to align x and y terms for easier solution. The equations are rewritten with the second equation multiplied by 3 to match the y terms, allowing for the cancellation of y and solving for x. Once x is found, y is determined by substituting the value of x back into one of the original equations. The solution to the system is given as (6,4).

05:05
🔍 Using Substitution with Decimals in Equations

The second paragraph explores the substitution method for solving equations with decimals. It presents a system of equations with Y isolated on one side in both equations, suggesting substitution as the preferred method. The equations are 0.2x + 0.4 = Y and 0.3x + 0.5Y = 3.4. The substitution of Y with 0.2x + 0.4 into the second equation leads to a new equation without fractions. To eliminate decimals, all terms are multiplied by 10, transforming the equation into a simpler form of 4x + 2 = 34. Solving for x gives the value of 8. This value is then substituted back into the equation for Y, yielding a final solution of (8,2).

Mindmap
Keywords
💡Equations
Equations are mathematical statements that assert the equality of two expressions. In the context of the video, equations are used to represent relationships between variables, such as 'x' and 'y'. The video focuses on solving systems of linear equations, which are fundamental in algebra and have wide applications in various fields.
💡Fractions
Fractions are representations of parts of a whole, typically expressed as a ratio of two integers. In the script, fractions are part of the equations that need to be solved. The process of clearing fractions, as mentioned in the video, simplifies the equations and makes them easier to work with, which is a common technique in algebra.
💡Decimals
Decimals are a way of representing numbers using a base-10 system, which includes a fractional part. The video script mentions decimals in the context of equations and suggests multiplying by a power of 10 to eliminate them, converting the equation into a simpler whole number form for easier calculation.
💡Elimination Method
The elimination method is a technique used to solve systems of equations by adding or subtracting equations to eliminate one variable, making it possible to solve for the other. In the video, this method is used when the 'y' variables align, allowing for their cancellation and simplifying the process of finding the value of 'x'.
💡Substitution
Substitution is another method for solving systems of equations where one variable is expressed in terms of the other and then substituted into the second equation. The video recommends using substitution when one of the variables is isolated, as it simplifies the process and directly leads to the solution for the other variable.
💡Variables
Variables are symbols, often letters like 'x' and 'y', used to represent unknown quantities in equations. The script discusses solving for these variables using different methods, emphasizing the importance of understanding how to manipulate and solve equations involving variables.
💡Multiplication
Multiplication is a fundamental arithmetic operation that combines groups of equal quantity. In the context of the video, multiplication is used to clear fractions and decimals from equations, which is a strategic move to simplify the equations before applying methods like elimination or substitution.
💡Solving
Solving refers to the process of finding the values of variables that make an equation true. The video script provides a step-by-step guide on how to solve equations, emphasizing the importance of this skill in mathematics and its applications.
💡Alignment
In the context of solving systems of equations, alignment refers to the arrangement of terms in such a way that when equations are combined, certain variables cancel out. The video mentions that when 'x' and 'y' are aligned, it's easier to use the elimination method.
💡Distributive Property
The distributive property is a fundamental property in algebra that allows you to multiply a term by a sum by multiplying the term by each addend separately and then summing the products. In the script, this property is used when distributing '0.5' across '0.2x + 0.4' to simplify the equation.
💡Solution
A solution in the context of equations is the set of values for the variables that satisfy the equation(s). The video script concludes with finding the solution to the given equations, which is represented as a pair of values for 'x' and 'y' that make the equations true.
Highlights

Solving equations with fractions by clearing them to simplify the process.

Multiplying the first equation by 3 to eliminate fractions.

Transforming 1/3 * 3 into 1 to simplify the equation.

Multiplying the second equation by 4 to clear fractions.

Using the elimination method when x and y are aligned.

Multiplying the second equation by 3 to cancel Y variables.

Adding equations to cancel Y variables and solve for X.

Finding the value of X to be 6 by dividing 42 by 7.

Subtracting 6 from both sides to isolate Y.

Solving for Y to be 4 by dividing 12 by 3.

The solution to the system of equations is (6, 4).

Choosing substitution over elimination when Y is isolated.

Substituting Y with 0.2X + 0.4 in the second equation.

Distributing 0.5 to eliminate decimals by multiplying through by 10.

Solving the modified equation to find X equals 8.

Plugging the value of X into the first equation to find Y.

Calculating Y to be 2 by substituting X and performing arithmetic.

The final answer to the system of equations is (2, 8).

Transcripts

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Equation SolvingFractionsDecimalsElimination MethodSubstitution MethodMathematicsAlgebraEducationalProblem SolvingTutorial