Subtracting 15 from Both Sides

Oct 13, 2025 by windailys.com 9 views

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will walk you through the step-by-step process of solving a linear equation, using the given equation as an example: $4 - 2(x + 7) = 3(x + 5)$ . We will break down the solution into manageable steps, making it easy to understand and follow.

Step 1: Distribute

The first step in solving the equation is to distribute the terms inside the parentheses. This means multiplying each term inside the parentheses by the coefficient outside the parentheses.

Distributing the Terms

To distribute the terms, we multiply each term inside the parentheses by the coefficient outside the parentheses.

$-2(x + 7)$ becomes $-2x - 14$
$3(x + 5)$ becomes $3x + 15$

The equation now becomes:

$4 - 2x - 14 = 3x + 15$

Simplifying the Equation

Now that we have distributed the terms, we can simplify the equation by combining like terms.

Combine the constant terms: $4 - 14 = -10$
Combine the x terms: $-2x = -2x$

The equation now becomes:

$-10 = 3x + 15$

Isolating the Variable

Our goal is to isolate the variable x. To do this, we need to get all the x terms on one side of the equation and the constant terms on the other side.

Subtracting 15 from Both Sides

To isolate the x term, we subtract 15 from both sides of the equation.

$-10 - 15 = 3x + 15 - 15$

This simplifies to:

$-25 = 3x$

Dividing Both Sides by 3

Finally, we divide both sides of the equation by 3 to solve for x.

$\frac{-25}{3} = \frac{3x}{3}$

This simplifies to:

$x = -\frac{25}{3}$

Conclusion

And that's it! We have successfully solved the linear equation using the step-by-step process outlined above. By distributing the terms, simplifying the equation, isolating the variable, and solving for x, we arrived at the solution: $x = -\frac{25}{3}$ .

Step 2: Combine Terms

Combining terms is an essential step in solving linear equations. When we combine terms, we add or subtract like terms to simplify the equation.

Combining Like Terms

To combine like terms, we add or subtract the coefficients of the like terms.

Combine the constant terms: $4 - 14 = -10$
Combine the x terms: $-2x = -2x$

The equation now becomes:

$-10 = 3x + 15$

Isolating the Variable

Our goal is to isolate the variable x. To do this, we need to get all the x terms on one side of the equation and the constant terms on the other side.

Subtracting 15 from Both Sides

To isolate the x term, we subtract 15 from both sides of the equation.

$-10 - 15 = 3x + 15 - 15$

This simplifies to:

$-25 = 3x$

Dividing Both Sides by 3

Finally, we divide both sides of the equation by 3 to solve for x.

$\frac{-25}{3} = \frac{3x}{3}$

This simplifies to:

$x = -\frac{25}{3}$

Conclusion

And that's it! We have successfully solved the linear equation using the step-by-step process outlined above. By combining terms, simplifying the equation, isolating the variable, and solving for x, we arrived at the solution: $x = -\frac{25}{3}$ .

Step 3: Apply Properties

Applying properties is an essential step in solving linear equations. When we apply properties, we use the rules of arithmetic to simplify the equation.

Adding, Subtracting, Multiplying, and Dividing

To apply properties, we use the rules of arithmetic to simplify the equation.

Add: $a + b = b + a$
Subtract: $a - b = b - a$
Multiply: $a \cdot b = b \cdot a$
Divide: $\frac{a}{b} = \frac{b}{a}$

The equation now becomes:

$4 - 2x - 14 = 3x + 15$

Simplifying the Equation

Now that we have applied the properties, we can simplify the equation by combining like terms.

Combine the constant terms: $4 - 14 = -10$
Combine the x terms: $-2x = -2x$

The equation now becomes:

$-10 = 3x + 15$

Isolating the Variable

Our goal is to isolate the variable x. To do this, we need to get all the x terms on one side of the equation and the constant terms on the other side.

Subtracting 15 from Both Sides

To isolate the x term, we subtract 15 from both sides of the equation.

$-10 - 15 = 3x + 15 - 15$

This simplifies to:

$-25 = 3x$

Dividing Both Sides by 3

Finally, we divide both sides of the equation by 3 to solve for x.

$\frac{-25}{3} = \frac{3x}{3}$

This simplifies to:

$x = -\frac{25}{3}$

Conclusion

And that's it! We have successfully solved the linear equation using the step-by-step process outlined above. By applying properties, simplifying the equation, isolating the variable, and solving for x, we arrived at the solution: $x = -\frac{25}{3}$ .

Step 4: To Start Over

Sometimes, we may need to start over and re-evaluate the equation. This can happen when we make a mistake or when the equation is too complex.

Resetting the Equation

To start over, we reset the equation to its original form.

$4 - 2(x + 7) = 3(x + 5)$

Distributing the Terms

To distribute the terms, we multiply each term inside the parentheses by the coefficient outside the parentheses.

$-2(x + 7)$ becomes $-2x - 14$
$3(x + 5)$ becomes $3x + 15$

The equation now becomes:

$4 - 2x - 14 = 3x + 15$

Simplifying the Equation

Now that we have distributed the terms, we can simplify the equation by combining like terms.

Combine the constant terms: $4 - 14 = -10$
Combine the x terms: $-2x = -2x$

The equation now becomes:

$-10 = 3x + 15$

Isolating the Variable

Our goal is to isolate the variable x. To do this, we need to get all the x terms on one side of the equation and the constant terms on the other side.

Subtracting 15 from Both Sides

To isolate the x term, we subtract 15 from both sides of the equation.

$-10 - 15 = 3x + 15 - 15$

This simplifies to:

$-25 = 3x$

Dividing Both Sides by 3

Finally, we divide both sides of the equation by 3 to solve for x.

$\frac{-25}{3} = \frac{3x}{3}$

This simplifies to:

$x = -\frac{25}{3}$

Conclusion

And that's it! We have successfully solved the linear equation using the step-by-step process outlined above. By starting over, distributing the terms, simplifying the equation, isolating the variable, and solving for x, we arrived at the solution: $x = -\frac{25}{3}$ .

Conclusion

Introduction

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving them. In this article, we will provide a Q&A guide to help you understand the process 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 (usually x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants.

Q: What are the steps involved in solving a linear equation?

A: The steps involved in solving a linear equation are:

Distribute the terms inside the parentheses.
Simplify the equation by combining like terms.
Isolate the variable by getting all the x terms on one side of the equation and the constant terms on the other side.
Solve for x by dividing both sides of the equation by the coefficient of x.

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 (usually x) is 1, while a quadratic equation is an equation in which the highest power of the variable (usually x) is 2. For example, 2x + 3 = 5 is a linear equation, while x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I distribute the terms inside the parentheses?

A: To distribute the terms inside the parentheses, you multiply each term inside the parentheses by the coefficient outside the parentheses. For example, in the equation 2(x + 3), you would multiply 2 by x and 2 by 3, resulting in 2x + 6.

Q: What is the difference between combining like terms and simplifying the equation?

A: Combining like terms involves adding or subtracting the coefficients of the like terms, while simplifying the equation involves reducing the equation to its simplest form by combining like terms and eliminating any unnecessary terms.

Q: How do I isolate the variable?

A: To isolate the variable, you need to get all the x terms on one side of the equation and the constant terms on the other side. You can do this by adding or subtracting the same value to both sides of the equation.

Q: What is the final step in solving a linear equation?

A: The final step in solving a linear equation is to solve for x by dividing both sides of the equation by the coefficient of x.

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

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

Not distributing the terms inside the parentheses
Not combining like terms
Not isolating the variable
Not solving for x
Making arithmetic errors

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by working through examples and exercises in a textbook or online resource. You can also try solving linear equations on your own by creating your own examples and solving them.

Conclusion

Solving linear equations is a fundamental concept in mathematics, and it's essential to understand the steps involved in solving them. By following the steps outlined in this article, you can successfully solve linear equations and arrive at the solution. Remember to distribute the terms, simplify the equation, isolate the variable, and solve for x. With practice and patience, you will become proficient in solving linear equations and tackle even the most complex equations with ease.