Solve For { X $} . . . { 5x = \}

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Introduction to Solving Equations

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to isolate variables to find their values. In this article, we'll focus on solving linear equations, specifically the equation $5x = 25$. We'll break down the steps to solve for $x$ and provide examples to help you understand the process.

Understanding the Equation

The equation $5x = 25$ is a linear equation, where $x$ is the variable, and $5$ is the coefficient. The equation states that the product of $5$ and $x$ is equal to $25$. Our goal is to find the value of $x$ that satisfies this equation.

Step 1: Divide Both Sides by 5

To solve for $x$, we need to isolate the variable. We can do this by dividing both sides of the equation by $5$. This will cancel out the coefficient $5$ and leave us with the value of $x$.

\frac{5x}{5} = \frac{25}{5}

Step 2: Simplify the Equation

After dividing both sides by $5$, we get:

x = 5

This is the value of $x$ that satisfies the equation $5x = 25$.

Example 1: Solving a Simple Equation

Let's consider another example: $3x = 12$. We can solve for $x$ by dividing both sides by $3$.

\frac{3x}{3} = \frac{12}{3}

This simplifies to:

x = 4

Example 2: Solving an Equation with a Larger Coefficient

Now, let's consider an equation with a larger coefficient: $7x = 49$. We can solve for $x$ by dividing both sides by $7$.

\frac{7x}{7} = \frac{49}{7}

This simplifies to:

x = 7

Tips and Tricks

Here are some tips and tricks to help you solve equations:

• Always check your work by plugging the solution back into the original equation.
• Make sure to follow the order of operations (PEMDAS) when simplifying the equation.
• If you're dealing with a fraction, make sure to simplify it before solving for the variable.

Conclusion

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to isolate variables to find their values. By following the steps outlined in this article, you'll be able to solve linear equations with ease. Remember to check your work, follow the order of operations, and simplify fractions before solving for the variable.

Frequently Asked Questions

• Q: What is the value of $x$ in the equation $5x = 25$? A: The value of $x$ is $5$.
• Q: How do I solve an equation with a larger coefficient? A: To solve an equation with a larger coefficient, divide both sides by the coefficient.
• Q: What are some tips and tricks for solving equations? A: Always check your work, follow the order of operations, and simplify fractions before solving for the variable.

Further Reading

If you're interested in learning more about solving equations, here are some additional resources:

• Khan Academy: Solving Linear Equations
• Mathway: Solving Equations
• Wolfram Alpha: Solving Equations

References

• [1] "Algebra" by Michael Artin
• [2] "Mathematics for the Nonmathematician" by Morris Kline
• [3] "Solving Equations" by Math Open Reference

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources.

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Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to isolate variables to find their values. In this article, we'll provide a comprehensive Q&A section to help you understand the process of solving equations.

Q: What is the first step in solving an equation?

A: The first step in solving an equation is to isolate the variable. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Q: How do I solve an equation with a variable on both sides?

A: To solve an equation with a variable on both sides, you need to get all the variables on one side of the equation and the constants on the other side. This can be done by adding or subtracting the same value to both sides of the equation.

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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula will give you two solutions for the equation.

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 simplifying an expression. 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 simplify a fraction?

A: To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and denominator and divide both numbers by the GCD.

Q: What is the difference between a rational equation and an irrational equation?

A: A rational equation is an equation in which the variable is in the numerator or denominator of a fraction, while an irrational equation is an equation in which the variable is not in the numerator or denominator of a fraction.

Q: How do I solve a rational equation?

A: To solve a rational equation, you need to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What is the difference between a system of equations and a single equation?

A: A system of equations is a set of two or more equations that are solved simultaneously, while a single equation is a single equation that is solved independently.

Q: How do I solve a system of equations?

A: To solve a system of equations, you need to use substitution or elimination to find the values of the variables that satisfy both equations.

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

A: A linear inequality is an inequality in which the highest power of the variable is 1, while a linear equation is an equation in which the highest power of the variable is 1.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable and determine the direction of the inequality.

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

A: A quadratic inequality is an inequality in which the highest power of the variable is 2, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and determine the intervals in which the inequality is true.

Q: What is the difference between a polynomial inequality and a polynomial equation?

A: A polynomial inequality is an inequality in which the highest power of the variable is a positive integer, while a polynomial equation is an equation in which the highest power of the variable is a positive integer.

Q: How do I solve a polynomial inequality?

A: To solve a polynomial inequality, you need to factor the polynomial expression and determine the intervals in which the inequality is true.

Conclusion

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to isolate variables to find their values. By following the steps outlined in this article, you'll be able to solve linear equations, quadratic equations, rational equations, and more. Remember to check your work, follow the order of operations, and simplify fractions before solving for the variable.

Frequently Asked Questions

• Q: What is the value of $x$ in the equation $5x = 25$? A: The value of $x$ is $5$.
• Q: How do I solve an equation with a larger coefficient? A: To solve an equation with a larger coefficient, divide both sides by the coefficient.
• Q: What are some tips and tricks for solving equations? A: Always check your work, follow the order of operations, and simplify fractions before solving for the variable.

Further Reading

If you're interested in learning more about solving equations, here are some additional resources:

• Khan Academy: Solving Linear Equations
• Mathway: Solving Equations
• Wolfram Alpha: Solving Equations

References

• [1] "Algebra" by Michael Artin
• [2] "Mathematics for the Nonmathematician" by Morris Kline
• [3] "Solving Equations" by Math Open Reference