Substitution Method For Solving Systems of Linear Equations, 2 and 3 Variables, Algebra 2

The Organic Chemistry Tutor
5 May 201711:36
EducationalLearning
32 Likes 10 Comments

TLDRThis instructional video demonstrates the substitution method for solving systems of linear equations. It begins with a simple example involving two equations with two variables, guiding viewers through the process of substitution and simplification to find the values of x and y. The video then tackles more complex scenarios, including equations with fractions and multiple variables, showing how to eliminate fractions and systematically solve for each variable. Each example is followed by a step-by-step explanation, making the content accessible for viewers to understand and apply the substitution method to various problems.

Takeaways
  • 📚 The video focuses on solving systems of linear equations using the substitution method.
  • 🔍 The first example involves two equations with 'y' expressed in terms of 'x', which are set equal to each other to find a single variable equation.
  • 📝 In the first example, by substituting and simplifying, it's found that x = 3 and subsequently y = 14, resulting in the solution (3,14).
  • 📈 The second example deals with equations involving fractions, where the least common multiple (LCM) is used to eliminate the fractions before solving.
  • 🧩 For the fraction-based example, after clearing the fractions and simplifying, it's determined that x = 6 and y = 0, leading to the solution (6,0).
  • 🔑 The third example provides a system with two equations and demonstrates substituting 'y' to find x = 2 and y = 3, with the solution (2,3).
  • 🔄 In the fourth example, 'y' is again substituted into the first equation, resulting in x = 4 and y = 3, giving the solution (4,3).
  • 📐 The final example increases complexity with three equations and variables, where 'z' and 'y' are substituted in terms of 'x' to find x = 1, y = 2, and z = 3, resulting in (1,2,3).
  • 📘 The video script provides step-by-step instructions on how to substitute and solve for variables in systems of linear equations.
  • 👍 The method demonstrated is effective for solving systems of equations, whether they involve two or three variables, and with or without fractions.
Q & A

  • What is the substitution method for solving a system of linear equations?

    -The substitution method involves replacing one variable in a set of equations with an expression that represents its value in terms of the other variables, allowing you to solve for one variable at a time.

  • How do you find the value of x and y from the equations y = 2x + 8 and y = 5x - 1?

    -You set the two expressions for y equal to each other (5x - 1 = 2x + 8) and solve for x. Once x is found, you substitute it back into either original equation to find the value of y.

  • What is the solution to the system of equations y = 2x + 8 and y = 5x - 1?

    -The solution is the ordered pair (x, y) = (3, 14), found by substituting and solving the equations as described.

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

    -You should eliminate fractions by finding the least common multiple (LCM) of the denominators and multiplying every term in the equation by this LCM to clear the fractions.

  • What is the least common multiple (LCM) between 2, 3, and 4, and why is it used?

    -The LCM between 2, 3, and 4 is 12. It is used to eliminate fractions by ensuring that every term in the equation can be multiplied by this number without leaving any fractions.

  • How do you solve the equation 1/4x - 3/2 = 2/3x - 4 after clearing the fractions?

    -After multiplying through by 12 to clear the fractions, you get 3x - 18 = 8x - 48. You then solve for x by combining like terms and isolating x on one side of the equation.

  • What is the solution to the system of equations y = 2/3x - 4 and y = 1/4x - 3/2?

    -The solution is the ordered pair (x, y) = (6, 0), found by substituting and solving the equations after clearing the fractions.

  • How do you approach a system of equations with three variables?

    -You can use the substitution method by expressing two of the variables in terms of the third and then solving the resulting equations step by step.

  • What is the solution to the system of equations x + y + z = 6, z = x + 2, and y = 2x?

    -The solution is the ordered triple (x, y, z) = (1, 2, 3), found by substituting the expressions for y and z into the first equation and solving for x, then finding y and z.

  • How do you solve the system of equations 5x - 4y = 8 and y = 1/2x + 1?

    -You substitute y in the first equation with 1/2x + 1, then solve for x. After finding x, you substitute it back into the equation for y to find its value.

  • What is the solution to the system of equations 5x - 4y = 8 and y = 1/2x + 1?

    -The solution is the ordered pair (x, y) = (4, 3), found by substituting and solving the equations as described.

Outlines
00:00
📚 Introduction to Solving Systems of Linear Equations by Substitution

This paragraph introduces the concept of solving systems of linear equations using the substitution method. The video begins with two equations involving 'x' and 'y', and demonstrates the process of substitution by replacing 'y' with its equivalent expression from the second equation. The goal is to simplify the system to a single equation with one variable, which can then be solved. The example provided involves basic algebraic manipulation, such as adding and subtracting terms, and dividing by a coefficient to isolate the variable. The solution for 'x' is found, and then substituted back into one of the original equations to find the corresponding value of 'y'. The final answer is presented as an ordered pair (x, y).

05:14
🔍 Solving Systems with Fractions and Multiple Examples

The second paragraph continues the theme of substitution but introduces fractions into the equations. The video script guides through the process of eliminating fractions by finding the least common multiple (LCM) of the denominators, which in this case is 12. Each term of the equation is then multiplied by 12 to clear the fractions. The resulting equation is simplified by combining like terms and isolating the variable 'x'. The solution for 'x' is found and substituted back into one of the original equations to solve for 'y'. The process is repeated with additional examples, each time emphasizing the substitution method and the algebraic steps required to find the solution for the variables involved. The solutions are presented as ordered pairs, demonstrating the successful application of the substitution method.

10:15
📘 Advanced Substitution with Three Variables

The final paragraph of the script tackles a more complex system involving three variables (x, y, z). The approach remains the substitution method, but with an additional variable, the process requires careful tracking of each substitution. The script shows how to replace 'z' and 'y' in the first equation with their respective expressions in terms of 'x'. This substitution leads to an equation with a single variable, which is then solved for 'x'. Once 'x' is determined, it is used to find the values of 'y' and 'z' by substituting back into their respective equations. The solution to this system is presented as an ordered triplet (x, y, z), illustrating the successful application of the substitution method to a more complex scenario.

Mindmap
Keywords
💡Substitution method
The substitution method is a technique used to solve systems of linear equations by replacing one variable with an expression involving the other variables. In the video, this method is the primary focus for finding the values of x and y in given equations. For example, in the first problem, 'y = 2x + 8' and 'y = 5x - 1' are set equal to each other, allowing 'y' to be substituted with '5x - 1', leading to the equation '5x - 1 = 2x + 8' which is then solved for 'x'.
💡System of linear equations
A system of linear equations refers to a collection of two or more equations involving the same set of variables. The video script discusses solving such systems using the substitution method. For instance, the script presents a system with two equations 'y = 2x + 8' and 'y = 5x - 1', which are solved together to find the values of 'x' and 'y'.
💡Variable
In the context of the video, a variable represents an unknown quantity that can take on different values. The script involves finding the values of variables 'x' and 'y'. The term is used in various equations throughout the video, such as in 'y = 2x + 8' where 'x' and 'y' are the variables whose values are being determined.
💡Least common multiple (LCM)
The least common multiple is the smallest number that is a multiple of two or more given numbers. In the video, when dealing with equations involving fractions, the LCM is used to eliminate the fractions by multiplying each term by the LCM to clear the denominators. For example, the script mentions finding the LCM of 2, 3, and 4 to solve the equation 'y = (2/3)x - 4'.
💡Ordered pair
An ordered pair in mathematics is a pair of numbers that are used to define a point in a coordinate system. In the video, once the values of 'x' and 'y' are found, they are written as ordered pairs to represent the solution to the system of equations. For instance, the solution to the first problem is represented as '(3, 14)'.
💡Distribute
To distribute in algebra means to multiply each term inside a set of parentheses by a number outside the parentheses and then add the results. In the script, distribution is used when solving the equation '2x + 5(3x - 3) = 19', where the '5' is distributed across '3x - 3' to get '15x - 15'.
💡Combine like terms
Combining like terms is the process of adding or subtracting terms in an equation that have the same variable raised to the same power. In the video, this is done in various examples, such as combining '2x' and '15x' to get '17x' in the equation '2x + 15x = 34'.
💡Equation
An equation is a mathematical statement that asserts the equality of two expressions. The video is centered around solving equations, specifically systems of linear equations. For example, the script starts with the equation 'y = 2x + 8' and another equation 'y = 5x - 1', which are set equal to each other to form a system.
💡Fraction
A fraction is a numerical expression that represents part of a whole, containing a numerator and a denominator. In the video, fractions are present in some of the equations, such as 'y = (2/3)x - 4', and methods to eliminate them, like finding the LCM, are discussed.
💡Solve
To solve in the context of mathematics means to find the value or values of the variables that make an equation true. The entire video is about solving systems of linear equations using the substitution method. For example, the script guides the viewer through the process of solving for 'x' and 'y' in the equation '5x - 1 = 2x + 8'.
Highlights

Introduction to solving a system of linear equations using the substitution method.

Problem setup with two equations involving y: y = 2x + 8 and y = 5x - 1.

Substitution of y with 5x - 1 to create a single equation with one variable.

Solving the equation 5x - 1 = 2x + 8 by simplifying and isolating x.

Finding x = 3 by solving the simplified equation 3x = 9.

Substituting x = 3 back into the original equations to find y.

Calculation of y = 14 using the value of x.

Presenting the solution as an ordered pair (x, y) = (3, 14).

Introduction to a second example with fractional equations.

Setting up the equation with fractions: y = 2/3x - 4 and y = 1/4x - 3/2.

Elimination of fractions by finding the least common multiple (LCM) of denominators.

Multiplying through by 12 to clear fractions and simplify the equation.

Solving for x after simplifying the equation to -5x = -30.

Finding x = 6 by dividing both sides by -5.

Substituting x = 6 into the original equation to solve for y.

Calculation of y = 0 using the value of x.

Presenting the solution as an ordered pair (x, y) = (6, 0).

Introduction to a problem with three variables and equations.

Setting up the system of equations with x, y, and z: x + y + z = 6, z = x + 2, y = 2x.

Substituting z and y in the first equation to solve for x.

Finding x = 1 by solving the equation 4x + 2 = 6.

Substituting x = 1 to find y and z.

Calculation of y = 2 and z = 3 using x = 1.

Presenting the solution as an ordered triple (x, y, z) = (1, 2, 3).

Transcripts

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