Solve Each Equation.1) { -3x - X = -24$}$2) { -5x - 2x = 14$}$3) { -2 = -3x + 4x$}$4) ${$7 = -6n + 7n$}$5) ${$7(7 - 7m) = 343$}$6) { -1 + 4(4n + 3) = -101$}$7) ${$2(1 - 5p) = 82$}$8)

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will guide you through the process of solving linear equations, using a step-by-step approach. We will cover various types of linear equations, including simple equations, equations with variables on both sides, and equations with parentheses.

Solving Simple Linear Equations

A simple linear equation is an equation that can be written in the form ax = b, where a and b are constants, and x is the variable. To solve a simple linear equation, we need to isolate the variable x.

Example 1: Solving the Equation -3x - x = -24

  • Step 1: Combine like terms on the left-hand side of the equation. −3x−x=−4x{-3x - x = -4x}
  • Step 2: Add 24 to both sides of the equation to isolate the term with the variable. −4x+24=−24+24{-4x + 24 = -24 + 24}
  • Step 3: Simplify the equation. −4x=−24{-4x = -24}
  • Step 4: Divide both sides of the equation by -4 to solve for x. x=−24−4{x = \frac{-24}{-4}}
  • Step 5: Simplify the fraction. x=6{x = 6}

Example 2: Solving the Equation -5x - 2x = 14

  • Step 1: Combine like terms on the left-hand side of the equation. −5x−2x=−7x{-5x - 2x = -7x}
  • Step 2: Add 14 to both sides of the equation to isolate the term with the variable. −7x+14=14+14{-7x + 14 = 14 + 14}
  • Step 3: Simplify the equation. −7x=28{-7x = 28}
  • Step 4: Divide both sides of the equation by -7 to solve for x. x=28−7{x = \frac{28}{-7}}
  • Step 5: Simplify the fraction. x=−4{x = -4}

Solving Linear Equations with Variables on Both Sides

A linear equation with variables on both sides is an equation that can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables. To solve a linear equation with variables on both sides, we need to isolate one of the variables.

Example 3: Solving the Equation -2 = -3x + 4x

  • Step 1: Combine like terms on the right-hand side of the equation. −2=x{-2 = x}
  • Step 2: The equation is already solved for x.

Example 4: Solving the Equation 7 = -6n + 7n

  • Step 1: Combine like terms on the right-hand side of the equation. 7=n{7 = n}
  • Step 2: The equation is already solved for n.

Solving Linear Equations with Parentheses

A linear equation with parentheses is an equation that contains parentheses in the expression. To solve a linear equation with parentheses, we need to follow the order of operations (PEMDAS) and simplify the expression inside the parentheses.

Example 5: Solving the Equation 7(7 - 7m) = 343

  • Step 1: Distribute the 7 to the terms inside the parentheses. 49−49m=343{49 - 49m = 343}
  • Step 2: Subtract 49 from both sides of the equation to isolate the term with the variable. −49m=294{-49m = 294}
  • Step 3: Divide both sides of the equation by -49 to solve for m. m=294−49{m = \frac{294}{-49}}
  • Step 4: Simplify the fraction. m=−6{m = -6}

Example 6: Solving the Equation -1 + 4(4n + 3) = -101

  • Step 1: Distribute the 4 to the terms inside the parentheses. −1+16n+12=−101{-1 + 16n + 12 = -101}
  • Step 2: Combine like terms on the left-hand side of the equation. 16n+11=−101{16n + 11 = -101}
  • Step 3: Subtract 11 from both sides of the equation to isolate the term with the variable. 16n=−112{16n = -112}
  • Step 4: Divide both sides of the equation by 16 to solve for n. n=−11216{n = \frac{-112}{16}}
  • Step 5: Simplify the fraction. n=−7{n = -7}

Example 7: Solving the Equation 2(1 - 5p) = 82

  • Step 1: Distribute the 2 to the terms inside the parentheses. 2−10p=82{2 - 10p = 82}
  • Step 2: Subtract 2 from both sides of the equation to isolate the term with the variable. −10p=80{-10p = 80}
  • Step 3: Divide both sides of the equation by -10 to solve for p. p=80−10{p = \frac{80}{-10}}
  • Step 4: Simplify the fraction. p=−8{p = -8}

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will answer some of the most frequently asked questions about solving linear equations.

Q: What is a linear equation?

A linear equation is an equation that can be written in the form ax = b, where a and b are constants, and x is the variable.

Q: How do I solve a linear equation?

To solve a linear equation, you need to isolate the variable x. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: What is the order of operations (PEMDAS)?

The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The acronym PEMDAS stands for:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify an expression inside parentheses?

To simplify an expression inside parentheses, you need to follow the order of operations (PEMDAS). First, evaluate any exponential expressions, then any multiplication and division operations, and finally any addition and subtraction operations.

Q: What is the difference between a simple linear equation and a linear equation with variables on both sides?

A simple linear equation is an equation that can be written in the form ax = b, where a and b are constants, and x is the variable. A linear equation with variables on both sides is an equation that can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

Q: How do I solve a linear equation with variables on both sides?

To solve a linear equation with variables on both sides, you need to isolate one of the variables. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

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

A linear equation is an equation that can be written in the form ax = b, where a and b are constants, and x is the variable. A quadratic equation is an equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable.

Q: How do I solve a quadratic equation?

To solve a quadratic equation, you need to use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. This formula will give you two solutions for x.

Q: What are some common mistakes to avoid when solving linear equations?

Some common mistakes to avoid when solving linear equations include:

  • Not following the order of operations (PEMDAS)
  • Not simplifying expressions inside parentheses
  • Not isolating the variable x
  • Not checking your solutions for x

Conclusion

Solving linear equations requires a step-by-step approach, and it's essential to follow the order of operations (PEMDAS) and simplify the expression inside the parentheses. By mastering these skills, you'll be able to solve a wide range of linear equations and become proficient in algebra.

Additional Resources

If you're struggling to solve linear equations, there are many online resources available to help you. Some popular resources include:

  • Khan Academy: Khan Academy offers a comprehensive course on algebra, including linear equations.
  • Mathway: Mathway is an online math problem solver that can help you solve linear equations and other math problems.
  • Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve linear equations and other math problems.

By following these tips and using these resources, you'll be well on your way to becoming proficient in solving linear equations.