Solve For $x$:$76 = 5x - 14$Simplify Your Answer As Much As Possible. $ X = X = X = [/tex]

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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 focus on solving a specific linear equation, $76 = 5x - 14$, and provide a step-by-step guide on how to simplify the solution.

What are 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. Linear equations can be solved using various methods, including algebraic manipulation, graphing, and substitution.

The Equation to Solve

The equation we will be solving is $76 = 5x - 14$. This equation is a linear equation in one variable, $x$. Our goal is to isolate $x$ and find its value.

Step 1: Add 14 to Both Sides

To start solving the equation, we need to get rid of the negative term on the right-hand side. We can do this by adding 14 to both sides of the equation. This will give us:

76+14=5x−14+1476 + 14 = 5x - 14 + 14

Simplifying the left-hand side, we get:

90=5x90 = 5x

Step 2: Divide Both Sides by 5

Now that we have the equation $90 = 5x$, we need to isolate $x$. We can do this by dividing both sides of the equation by 5. This will give us:

905=5x5\frac{90}{5} = \frac{5x}{5}

Simplifying both sides, we get:

18=x18 = x

Conclusion

In this article, we solved the linear equation $76 = 5x - 14$ using algebraic manipulation. We added 14 to both sides of the equation to get rid of the negative term, and then divided both sides by 5 to isolate $x$. The final solution is $x = 18$.

Tips and Tricks

  • When solving linear equations, it's essential to follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
  • When adding or subtracting terms, make sure to add or subtract the coefficients of the terms, not the variables.
  • When dividing both sides of an equation by a term, make sure to divide both the coefficients and the variables by that term.

Real-World Applications

Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and distance.
  • Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
  • Economics: Linear equations are used to model economic systems, including supply and demand, and cost-benefit analysis.

Common Mistakes to Avoid

  • When solving linear equations, it's easy to make mistakes by adding or subtracting the wrong terms, or dividing both sides by the wrong term.
  • When working with fractions, make sure to simplify the fractions before solving the equation.
  • When using algebraic manipulation, make sure to follow the order of operations and simplify the equation step-by-step.

Conclusion

Introduction

In our previous article, we solved the linear equation $76 = 5x - 14$ using algebraic manipulation. In this article, we will provide a Q&A guide to help you better understand the concept of solving linear equations.

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 solve a linear equation?

A: To solve a linear equation, you need to isolate the variable(s) by performing algebraic operations such as addition, subtraction, multiplication, and division. You can use the following steps:

  1. Add or subtract the same value to both sides of the equation to get rid of the constant term.
  2. Multiply or divide both sides of the equation by the same value to get rid of the coefficient of the variable.
  3. Simplify the equation by combining like terms.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when solving an equation. The order of operations is:

  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 handle fractions in linear equations?

A: When working with fractions in linear equations, you need to follow the same steps as solving linear equations. However, you need to simplify the fractions before solving the equation. You can do this by:

  1. Finding the least common multiple (LCM) of the denominators.
  2. Multiplying both sides of the equation by the LCM.
  3. Simplifying the equation by combining like terms.

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

A: Some common mistakes to avoid when solving linear equations include:

  • Adding or subtracting the wrong terms.
  • Dividing both sides of the equation by the wrong term.
  • Not simplifying the equation before solving it.
  • Not following the order of operations.

Q: How do I apply linear equations to real-world problems?

A: Linear equations have numerous real-world applications, including:

  • Physics: Linear equations are used to describe the motion of objects, including velocity, acceleration, and distance.
  • Engineering: Linear equations are used to design and optimize systems, including electrical circuits, mechanical systems, and control systems.
  • Economics: Linear equations are used to model economic systems, including supply and demand, and cost-benefit analysis.

Q: What are some tips for solving linear equations?

A: Some tips for solving linear equations include:

  • Follow the order of operations.
  • Simplify the equation before solving it.
  • Use algebraic manipulation to isolate the variable(s).
  • Check your work by plugging the solution back into the original equation.

Conclusion

Solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, you can solve linear equations with ease. Remember to follow the order of operations, simplify the equation step-by-step, and avoid common mistakes. With practice and patience, you'll become proficient in solving linear equations and apply them to real-world problems.