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Systems of equations with substitution: -3x-4y=-2 & y=2x-5

Simple Guide: Solving Systems of Equations with Examples and Worksheets

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Simple Guide: Solving Systems of Equations with Examples and Worksheets
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Mahak Tiwari

@mahaktiwari_ikmn

·

70 Followers

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This document provides a comprehensive guide on solving systems of equations by substitution method. It covers key concepts, step-by-step procedures, and multiple example problems to illustrate the technique.

  • The substitution method is an algebraic approach for finding exact solutions to systems of linear equations
  • It involves expressing one variable in terms of another and substituting into the other equation
  • The document walks through several example problems of increasing complexity
  • Key concepts like consistent, inconsistent, and dependent systems are explained

2/17/2023

1493

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

View

Final Set of Practice Problems

This page contains the last set of practice problems for solving linear systems by substitution. These problems are designed to test and reinforce all the concepts and techniques covered in the previous sections.

The problems include a mix of: • Straightforward systems • Systems requiring algebraic manipulation • Systems with special solutions (no solution or infinite solutions)

Example: One problem asks to solve: y = -x + 2 -4x + y = 2 This system combines a pre-solved equation with one requiring substitution.

Detailed solutions are provided, allowing students to check their work and review the problem-solving process.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

View

Practice Problems: Solving by Elimination and Substitution

This final section provides mixed practice problems using both elimination and substitution methods.

Key features:

  • Problems are designed to reinforce both techniques.
  • Students must choose the most appropriate method for each system.
  • Detailed solutions are provided, demonstrating method selection and problem-solving steps.

Example: A problem is solved using substitution: y = 2x + 1 2x + y = 14

The solution process clearly shows why substitution is efficient for this particular system.

Highlight: This mixed practice helps students develop the skill of selecting the most efficient method for solving different types of systems of equations.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

View

Introduction to Solving Systems of Linear Equations by Substitution

This page introduces the substitution method for solving systems of linear equations. It explains that while graphing can provide approximate solutions, algebraic methods like substitution yield exact solutions.

The key steps of the substitution method are outlined:

  1. Express one variable in terms of the other
  2. Substitute that expression into the other equation
  3. Solve for the remaining variable
  4. Substitute back to find the other variable's value

Definition: The substitution method is an algebraic technique for finding the exact point of intersection of two linear equations by expressing one variable in terms of the other.

A detailed example is provided, showing how to solve the system: 2x + y = -11 y = 3x - 9

The solution process is explained step-by-step, resulting in the solution (2, -3).

Example: Substituting y = 3x - 9 into 2x + y = -11 yields 2x + (3x - 9) = -11. Solving this equation gives x = 2, which can then be used to find y = -3.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

View

Introduction to Elimination Method

The final page introduces the elimination method as another technique for solving systems of linear equations. This brief introduction sets the stage for the next section of the course.

Vocabulary: The elimination method, also known as addition method, involves adding or subtracting equations to eliminate one variable.

This page serves as a transition, indicating that students will learn an alternative method to substitution for solving linear systems in the upcoming lessons.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

View

Advanced Practice Problems for Linear Systems

This page presents more challenging practice problems for solving systems of linear equations by substitution. These problems require more complex algebraic manipulation and critical thinking.

Problem types include: • Systems with fractional coefficients • Systems requiring multiple substitutions • Systems with more complex algebraic expressions

Highlight: Some problems on this page may require students to combine the substitution method with other algebraic techniques for efficient solving.

Solutions are provided, demonstrating advanced problem-solving strategies and reinforcing the versatility of the substitution method for solving linear systems.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

View

Additional Examples of Solving Linear Systems by Substitution

This page provides more examples of solving linear systems by substitution, demonstrating different scenarios students may encounter.

Example 1 solves the system: 5x + y = 4 2x - 3y = 5

It shows how to solve for y in terms of x, substitute into the other equation, and solve for x and y.

Example 2 demonstrates an inconsistent system with no solution: 4x + 2y = 5 y = -2x + 1

Highlight: An inconsistent system of equations has no solution, which is revealed when substitution leads to a false equation like 2 = 5.

Example 3 shows a dependent system with infinitely many solutions: 6x - 2y = 4 y = 3x - 2

Vocabulary: A dependent system of equations has infinitely many solutions, occurring when substitution leads to a true equation for all values of the variables.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

View

Additional Practice Problems and Solutions

This page continues with more practice problems for solving linear systems by substitution. It includes a mix of problem types to reinforce students' skills.

Key problem types include: • Systems where one equation is already solved for a variable • Systems requiring algebraic manipulation before substitution • Systems resulting in no solution or infinite solutions

Example: One problem solves the system: x = 2y - 3 2x - 3y = -5 This requires substituting the expression for x into the second equation.

Detailed solutions are provided for each problem, showing the step-by-step process of using the substitution method.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

View

Practice Problems for Solving Linear Systems by Substitution

This page provides a set of practice problems for students to apply the substitution method for solving systems of linear equations. The problems range in difficulty and include various types of systems.

Some key problem types include: • Systems with whole number solutions • Systems with fractional solutions • Inconsistent systems (no solution) • Dependent systems (infinite solutions)

Example: One practice problem is to solve the system: 2x + 3y = 7 x = 2 This system has a straightforward solution of (2, 1).

The page includes solutions to odd-numbered problems, allowing students to check their work and understanding of the substitution method.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

View

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

View

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Subjects

/

Algebra 1

/

Systems of equations

/

Solving systems of equations with substitution

/

Systems of equations with substitution: -3x-4y=-2 & y=2x-5

Simple Guide: Solving Systems of Equations with Examples and Worksheets

user profile picture

Mahak Tiwari

@mahaktiwari_ikmn

·

70 Followers

Follow

This document provides a comprehensive guide on solving systems of equations by substitution method. It covers key concepts, step-by-step procedures, and multiple example problems to illustrate the technique.

  • The substitution method is an algebraic approach for finding exact solutions to systems of linear equations
  • It involves expressing one variable in terms of another and substituting into the other equation
  • The document walks through several example problems of increasing complexity
  • Key concepts like consistent, inconsistent, and dependent systems are explained

2/17/2023

1493

 

Algebra 1

63

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Final Set of Practice Problems

This page contains the last set of practice problems for solving linear systems by substitution. These problems are designed to test and reinforce all the concepts and techniques covered in the previous sections.

The problems include a mix of: • Straightforward systems • Systems requiring algebraic manipulation • Systems with special solutions (no solution or infinite solutions)

Example: One problem asks to solve: y = -x + 2 -4x + y = 2 This system combines a pre-solved equation with one requiring substitution.

Detailed solutions are provided, allowing students to check their work and review the problem-solving process.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Practice Problems: Solving by Elimination and Substitution

This final section provides mixed practice problems using both elimination and substitution methods.

Key features:

  • Problems are designed to reinforce both techniques.
  • Students must choose the most appropriate method for each system.
  • Detailed solutions are provided, demonstrating method selection and problem-solving steps.

Example: A problem is solved using substitution: y = 2x + 1 2x + y = 14

The solution process clearly shows why substitution is efficient for this particular system.

Highlight: This mixed practice helps students develop the skill of selecting the most efficient method for solving different types of systems of equations.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Introduction to Solving Systems of Linear Equations by Substitution

This page introduces the substitution method for solving systems of linear equations. It explains that while graphing can provide approximate solutions, algebraic methods like substitution yield exact solutions.

The key steps of the substitution method are outlined:

  1. Express one variable in terms of the other
  2. Substitute that expression into the other equation
  3. Solve for the remaining variable
  4. Substitute back to find the other variable's value

Definition: The substitution method is an algebraic technique for finding the exact point of intersection of two linear equations by expressing one variable in terms of the other.

A detailed example is provided, showing how to solve the system: 2x + y = -11 y = 3x - 9

The solution process is explained step-by-step, resulting in the solution (2, -3).

Example: Substituting y = 3x - 9 into 2x + y = -11 yields 2x + (3x - 9) = -11. Solving this equation gives x = 2, which can then be used to find y = -3.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Introduction to Elimination Method

The final page introduces the elimination method as another technique for solving systems of linear equations. This brief introduction sets the stage for the next section of the course.

Vocabulary: The elimination method, also known as addition method, involves adding or subtracting equations to eliminate one variable.

This page serves as a transition, indicating that students will learn an alternative method to substitution for solving linear systems in the upcoming lessons.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Advanced Practice Problems for Linear Systems

This page presents more challenging practice problems for solving systems of linear equations by substitution. These problems require more complex algebraic manipulation and critical thinking.

Problem types include: • Systems with fractional coefficients • Systems requiring multiple substitutions • Systems with more complex algebraic expressions

Highlight: Some problems on this page may require students to combine the substitution method with other algebraic techniques for efficient solving.

Solutions are provided, demonstrating advanced problem-solving strategies and reinforcing the versatility of the substitution method for solving linear systems.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Additional Examples of Solving Linear Systems by Substitution

This page provides more examples of solving linear systems by substitution, demonstrating different scenarios students may encounter.

Example 1 solves the system: 5x + y = 4 2x - 3y = 5

It shows how to solve for y in terms of x, substitute into the other equation, and solve for x and y.

Example 2 demonstrates an inconsistent system with no solution: 4x + 2y = 5 y = -2x + 1

Highlight: An inconsistent system of equations has no solution, which is revealed when substitution leads to a false equation like 2 = 5.

Example 3 shows a dependent system with infinitely many solutions: 6x - 2y = 4 y = 3x - 2

Vocabulary: A dependent system of equations has infinitely many solutions, occurring when substitution leads to a true equation for all values of the variables.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Additional Practice Problems and Solutions

This page continues with more practice problems for solving linear systems by substitution. It includes a mix of problem types to reinforce students' skills.

Key problem types include: • Systems where one equation is already solved for a variable • Systems requiring algebraic manipulation before substitution • Systems resulting in no solution or infinite solutions

Example: One problem solves the system: x = 2y - 3 2x - 3y = -5 This requires substituting the expression for x into the second equation.

Detailed solutions are provided for each problem, showing the step-by-step process of using the substitution method.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Practice Problems for Solving Linear Systems by Substitution

This page provides a set of practice problems for students to apply the substitution method for solving systems of linear equations. The problems range in difficulty and include various types of systems.

Some key problem types include: • Systems with whole number solutions • Systems with fractional solutions • Inconsistent systems (no solution) • Dependent systems (infinite solutions)

Example: One practice problem is to solve the system: 2x + 3y = 7 x = 2 This system has a straightforward solution of (2, 1).

The page includes solutions to odd-numbered problems, allowing students to check their work and understanding of the substitution method.

Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

Sign up to see the content. It's free!

Access to all documents

Improve your grades

Join milions of students

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Section 6.2: Solving Systems of Linear Equations (substitution) 3/25/18 1. Finding a graphical solution of a system of equations is based on

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Knowunity was a featured story by Apple and has consistently topped the app store charts within the education category in Germany, Italy, Poland, Switzerland and United Kingdom. Join Knowunity today and help millions of students around the world.

Ranked #1 Education App

Download in

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Download in

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Students use Knowunity

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Stefan S, iOS User

The application is very simple and well designed. So far I have found what I was looking for :D

SuSSan, iOS User

Love this App ❤️, I use it basically all the time whenever I'm studying