Sort Each Equation According To Whether It Has One Solution, Infinitely Many Solutions, Or No Solution.1. $5(x-2)=5x-7$2. $-3(x-4)=-3x+12$3. $4(x+1)=3x+4$4. $-2(x-3)=2x-6$5. $6(x+5)=6x+11$Infinitely Many

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Introduction

Linear equations are a fundamental concept in mathematics, and understanding their solutions is crucial for solving various mathematical problems. In this article, we will explore how to sort linear equations into three categories: one solution, infinitely many solutions, or no solution. We will examine five different equations and determine the type of solution for each.

Understanding Linear Equations

A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form:

ax + b = c

where a, b, and c are constants, and x is the variable.

Sorting Equations: One Solution, Infinitely Many Solutions, or No Solution

To sort an equation into one of the three categories, we need to follow these steps:

  1. Simplify the equation: Combine like terms and isolate the variable.
  2. Check for equality: Determine if the equation is true or false.
  3. Determine the solution: Based on the result, determine if the equation has one solution, infinitely many solutions, or no solution.

Equation 1: 5(x−2)=5x−75(x-2)=5x-7

Let's start by simplifying the equation:

5(x−2)=5x−75(x-2)=5x-7

Expand the left side:

5x−10=5x−75x-10=5x-7

Subtract 5x5x from both sides:

−10=−7-10=-7

This equation is false, so it has no solution.

Equation 2: −3(x−4)=−3x+12-3(x-4)=-3x+12

Simplify the equation:

−3(x−4)=−3x+12-3(x-4)=-3x+12

Expand the left side:

−3x+12=−3x+12-3x+12=-3x+12

Subtract −3x-3x from both sides:

12=1212=12

This equation is true, so it has infinitely many solutions.

Equation 3: 4(x+1)=3x+44(x+1)=3x+4

Simplify the equation:

4(x+1)=3x+44(x+1)=3x+4

Expand the left side:

4x+4=3x+44x+4=3x+4

Subtract 3x3x from both sides:

x=0x=0

This equation is true, so it has one solution.

Equation 4: −2(x−3)=2x−6-2(x-3)=2x-6

Simplify the equation:

−2(x−3)=2x−6-2(x-3)=2x-6

Expand the left side:

−2x+6=2x−6-2x+6=2x-6

Add 2x2x to both sides:

6=4x−66=4x-6

Add 6 to both sides:

12=4x12=4x

Divide both sides by 4:

3=x3=x

This equation is true, so it has one solution.

Equation 5: 6(x+5)=6x+116(x+5)=6x+11

Simplify the equation:

6(x+5)=6x+116(x+5)=6x+11

Expand the left side:

6x+30=6x+116x+30=6x+11

Subtract 6x6x from both sides:

30=1130=11

This equation is false, so it has no solution.

Conclusion

In this article, we have explored how to sort linear equations into three categories: one solution, infinitely many solutions, or no solution. We have examined five different equations and determined the type of solution for each. By following the steps outlined in this article, you can easily sort linear equations and understand their solutions.

Key Takeaways

  • Linear equations can be sorted into three categories: one solution, infinitely many solutions, or no solution.
  • To sort an equation, simplify it, check for equality, and determine the solution.
  • Infinitely many solutions occur when the equation is true for all values of the variable.
  • One solution occurs when the equation is true for a specific value of the variable.
  • No solution occurs when the equation is false for all values of the variable.

Further Reading

If you want to learn more about linear equations and their solutions, we recommend checking out the following resources:

  • Khan Academy: Linear Equations
  • Mathway: Linear Equations
  • Wolfram MathWorld: Linear Equations

Introduction

In our previous article, we explored how to sort linear equations into three categories: one solution, infinitely many solutions, or no solution. In this article, we will answer some frequently asked questions about linear equations and their solutions.

Q&A

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It can be written in the form:

ax + b = c

where a, b, and c are constants, and x is the variable.

Q: How do I simplify a linear equation?

A: To simplify a linear equation, combine like terms and isolate the variable. For example, consider the equation:

2x + 3 = 5

Subtract 3 from both sides:

2x = 2

Divide both sides by 2:

x = 1

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, consider the equation:

x^2 + 2x + 1 = 0

This is a quadratic equation because the highest power of x is 2.

Q: How do I determine the solution of a linear equation?

A: To determine the solution of a linear equation, follow these steps:

  1. Simplify the equation by combining like terms and isolating the variable.
  2. Check if the equation is true or false.
  3. If the equation is true, it has infinitely many solutions.
  4. If the equation is false, it has no solution.
  5. If the equation is true for a specific value of the variable, it has one solution.

Q: Can a linear equation have more than one solution?

A: No, a linear equation can only have one solution or infinitely many solutions. If a linear equation has a solution, it is a unique solution.

Q: Can a linear equation have no solution?

A: Yes, a linear equation can have no solution if it is false for all values of the variable.

Q: How do I graph a linear equation?

A: To graph a linear equation, follow these steps:

  1. Simplify the equation by combining like terms and isolating the variable.
  2. Determine the slope and y-intercept of the equation.
  3. Plot the y-intercept on the coordinate plane.
  4. Use the slope to plot additional points on the coordinate plane.
  5. Draw a line through the points to graph the equation.

Q: What is the importance of linear equations in real-life applications?

A: Linear equations are used in a wide range of real-life applications, including:

  • Physics: Linear equations are used to describe the motion of objects and the forces acting on them.
  • Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear equations are used to model economic systems and make predictions about economic trends.
  • Computer Science: Linear equations are used in algorithms and data structures to solve problems efficiently.

Conclusion

In this article, we have answered some frequently asked questions about linear equations and their solutions. We hope that this article has provided you with a better understanding of linear equations and their applications.

Key Takeaways

  • Linear equations can be sorted into three categories: one solution, infinitely many solutions, or no solution.
  • To determine the solution of a linear equation, simplify it, check for equality, and determine the solution.
  • Linear equations can be used to model real-life applications, such as physics, engineering, economics, and computer science.

Further Reading

If you want to learn more about linear equations and their solutions, we recommend checking out the following resources:

  • Khan Academy: Linear Equations
  • Mathway: Linear Equations
  • Wolfram MathWorld: Linear Equations

By following the steps outlined in this article and exploring the resources listed above, you can gain a deeper understanding of linear equations and their solutions.