Distributing the Negative 4

Oct 13, 2025 by magicalceylontours.com 32 views

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, $-4(8+y)=90$ , and provide a step-by-step guide on how to approach it.

What is a Linear Equation?

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 $-4(8+y)=90$

The given equation is $-4(8+y)=90$ . To solve this equation, we need to isolate the variable $y$ . The first step is to simplify the left-hand side of the equation by distributing the negative 4 to the terms inside the parentheses.

Distributing the Negative 4

When we distribute the negative 4 to the terms inside the parentheses, we get:

-4(8+y) = -32 - 4y

Now, the equation becomes:

-32 - 4y = 90

Adding 32 to Both Sides

To isolate the term with the variable, we need to get rid of the constant term on the left-hand side. We can do this by adding 32 to both sides of the equation.

-4y = 90 + 32

-4y = 122

Dividing Both Sides by -4

Now, we need to isolate the variable $y$ by dividing both sides of the equation by -4.

y = -\frac{122}{4}

y = -30.5

Conclusion

In this article, we solved the linear equation $-4(8+y)=90$ using algebraic manipulation. We simplified the left-hand side of the equation by distributing the negative 4, added 32 to both sides to isolate the term with the variable, and finally divided both sides by -4 to find the value of $y$ . The solution to the equation is $y = -30.5$ .

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS):

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

By following these steps and using algebraic manipulation, you can solve linear equations with ease.

Real-World Applications

Linear equations have numerous real-world applications in fields such as physics, engineering, economics, and computer science. For example, in physics, linear equations can be used to model the motion of objects, while in economics, they can be used to analyze the relationship between variables such as supply and demand.

Common Mistakes to Avoid

When solving linear equations, it's essential to avoid common mistakes such as:

Not following the order of operations (PEMDAS)
Not simplifying the left-hand side of the equation
Not isolating the variable
Not checking the solution for validity

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Conclusion

Introduction

In our previous article, we discussed how to solve linear equations using algebraic manipulation. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in 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 simplify a linear equation?

A: To simplify a linear equation, you need to follow the order of operations (PEMDAS):

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

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, you need to get rid of the constant term on the same side of the equation as the variable. You can do this 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(s) is 1, while a quadratic equation is an equation in which the highest power of the variable(s) is 2. For example, $x + 2 = 3$ is a linear equation, while $x^2 + 2x + 1 = 0$ is a quadratic equation.

Q: How do I solve a linear equation with fractions?

A: To solve a linear equation with fractions, you need to follow the same steps as solving a linear equation with integers. However, you may need to multiply both sides of the equation by the denominator to eliminate the fraction.

Q: What is the importance of checking the solution for validity?

A: Checking the solution for validity is crucial to ensure that the solution is correct. You can do this by plugging the solution back into the original equation and verifying that it is true.

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

A: Linear equations can be applied to a wide range of real-world problems, including physics, engineering, economics, and computer science. For example, in physics, linear equations can be used to model the motion of objects, while in economics, they can be used to analyze the relationship between variables such as supply and demand.

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

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

Not following the order of operations (PEMDAS)
Not simplifying the left-hand side of the equation
Not isolating the variable
Not checking the solution for validity

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the steps outlined in this article and using algebraic manipulation, you can solve linear equations with ease. Remember to follow the order of operations (PEMDAS), simplify the left-hand side of the equation, isolate the variable, and check the solution for validity. With practice and patience, you can become proficient in solving linear equations and apply them to real-world problems.

Additional Resources

For more information on solving linear equations, you can refer to the following resources:

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

Practice Problems

To practice solving linear equations, try the following problems:

Solve the equation $2x + 3 = 7$ .
Solve the equation $x - 2 = 5$ .
Solve the equation $3x + 2 = 11$ .

Answer Key

$x = 2$
$x = 7$
$x = 3$