Given { H(x) = -2x - 5 $}$, Solve For { X $}$ When { H(x) = 9 $}$.

Mar 9, 2025 by ADMIN 67 views

**Solving Linear Equations: A Step-by-Step Guide**

Introduction

In mathematics, solving linear equations is a fundamental concept that forms the basis of various mathematical operations. A linear equation is an equation in which the highest power of the variable(s) is 1. In this article, we will focus on solving linear equations of the form h(x) = mx + b, where m is the slope and b is the y-intercept. We will use the given equation h(x) = -2x - 5 to solve for x when h(x) = 9.

Understanding the Given Equation

The given equation is h(x) = -2x - 5. This equation represents a linear function with a slope of -2 and a y-intercept of -5. The slope of a linear function represents the rate of change of the function with respect to the variable x, while the y-intercept represents the value of the function when x is equal to 0.

Solving for x

To solve for x, we need to isolate the variable x on one side of the equation. We can do this by adding 5 to both sides of the equation, which will eliminate the constant term -5. This gives us:

h(x) = -2x - 5 h(x) + 5 = -2x

Next, we can add 2x to both sides of the equation to isolate the term -2x. This gives us:

h(x) + 5 + 2x = -2x + 2x h(x) + 5 + 2x = 0

Now, we can subtract 5 from both sides of the equation to eliminate the constant term 5. This gives us:

h(x) + 2x = -5

Finally, we can subtract h(x) from both sides of the equation to isolate the term 2x. This gives us:

2x = -5 - h(x)

Substituting the Value of h(x)

We are given that h(x) = 9. We can substitute this value into the equation 2x = -5 - h(x) to solve for x. This gives us:

2x = -5 - 9 2x = -14

Solving for x

To solve for x, we can divide both sides of the equation by 2. This gives us:

x = -14/2 x = -7

Conclusion

In this article, we solved the linear equation h(x) = -2x - 5 for x when h(x) = 9. We used the steps of adding 5 to both sides of the equation, adding 2x to both sides of the equation, subtracting 5 from both sides of the equation, and subtracting h(x) from both sides of the equation to isolate the term 2x. Finally, we substituted the value of h(x) into the equation and solved for x. The solution to the equation is x = -7.

Example Use Cases

Solving linear equations is a fundamental concept in mathematics that has numerous applications in various fields, including physics, engineering, economics, and computer science. Here are a few example use cases:

Physics: In physics, linear equations are used to describe the motion of objects under the influence of forces. For example, the equation h(x) = -2x - 5 can be used to describe the motion of an object under the influence of a constant force.
Engineering: In engineering, linear equations are used to design and optimize systems. For example, the equation h(x) = -2x - 5 can be used to design a system that minimizes the cost of production while maximizing the output.
Economics: In economics, linear equations are used to model the behavior of economic systems. For example, the equation h(x) = -2x - 5 can be used to model the demand for a product as a function of its price.

Tips and Tricks

Here are a few tips and tricks to help you solve linear equations:

Use the distributive property: The distributive property states that a(b + c) = ab + ac. This property can be used to simplify linear equations by distributing the coefficients to the terms inside the parentheses.
Use the commutative property: The commutative property states that a + b = b + a. This property can be used to rearrange the terms in a linear equation to make it easier to solve.
Use the associative property: The associative property states that (a + b) + c = a + (b + c). This property can be used to rearrange the terms in a linear equation to make it easier to solve.

Conclusion

Introduction

In our previous article, we discussed how to solve linear equations of the form h(x) = mx + b, where m is the slope and b is the y-intercept. We used the given equation h(x) = -2x - 5 to solve for x when h(x) = 9. In this article, we will provide a Q&A guide to help you understand and solve 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 h(x) = mx + b, where m is the slope and b is the y-intercept.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable x on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, and then dividing or multiplying both sides of the equation by the same non-zero value.

Q: What is the slope of a linear equation?

A: The slope of a linear equation is the coefficient of the variable x. It represents the rate of change of the function with respect to the variable x.

Q: What is the y-intercept of a linear equation?

A: The y-intercept of a linear equation is the value of the function when x is equal to 0. It is represented by the constant term b in the equation h(x) = mx + b.

Q: How do I find the value of x in a linear equation?

A: To find the value of x in a linear equation, you need to isolate the variable x on one side of the equation. You can do this by adding or subtracting the same value to both sides of the equation, and then dividing or multiplying both sides of the equation by the same non-zero value.

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, the equation h(x) = -2x - 5 is a linear equation, while the equation h(x) = x^2 + 2x + 1 is a quadratic equation.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to find the values of the variables that satisfy all the equations in the system. You can do this by using methods such as substitution, elimination, or graphing.

Q: What is the importance of solving linear equations?

A: Solving linear equations is an important concept in mathematics that has numerous applications in various fields, including physics, engineering, economics, and computer science. It helps us to model real-world problems, make predictions, and optimize systems.

Q: How do I practice solving linear equations?

A: You can practice solving linear equations by working on exercises and problems, such as those found in textbooks or online resources. You can also try solving real-world problems that involve linear equations.

Conclusion

In conclusion, solving linear equations is an important concept in mathematics that has numerous applications in various fields. By understanding the concepts of slope, y-intercept, and linear equations, you can solve linear equations and apply them to real-world problems. We hope this Q&A guide has helped you to understand and solve linear equations.

Example Problems

Here are a few example problems to help you practice solving linear equations:

Solve the equation h(x) = 2x + 3 for x when h(x) = 5.
Solve the equation h(x) = -x + 2 for x when h(x) = -3.
Solve the system of linear equations:

h(x) = 2x + 3 h(x) = -x + 2

for x and y.

Tips and Tricks

Here are a few tips and tricks to help you solve linear equations:

Use the distributive property: The distributive property states that a(b + c) = ab + ac. This property can be used to simplify linear equations by distributing the coefficients to the terms inside the parentheses.
Use the commutative property: The commutative property states that a + b = b + a. This property can be used to rearrange the terms in a linear equation to make it easier to solve.
Use the associative property: The associative property states that (a + b) + c = a + (b + c). This property can be used to rearrange the terms in a linear equation to make it easier to solve.