Solve For $w$.$-9w - 5 + 6w = 19$Simplify Your Answer As Much As Possible.$w =$

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 linear equation of the form $-9w - 5 + 6w = 19$ to isolate the variable $w$. We will break down the solution into manageable steps, making it easy to understand and follow along.

Understanding the Equation

The given equation is $-9w - 5 + 6w = 19$. Our goal is to isolate the variable $w$ on one side of the equation. To do this, we need to simplify the equation by combining like terms and performing algebraic operations.

Combining Like Terms

The first step in solving the equation is to combine like terms. In this case, we have two terms with the variable $w$: $-9w$ and $6w$. We can combine these terms by adding their coefficients.

# Combining like terms
w_coefficient = -9 + 6
print(w_coefficient)

The output of the code is $-3$. So, the equation becomes $-3w - 5 = 19$.

Isolating the Variable w

Now that we have combined like terms, we can isolate the variable $w$ by performing algebraic operations. We want to get rid of the constant term $-5$ on the left-hand side of the equation. To do this, we can add $5$ to both sides of the equation.

# Isolating the variable w
constant_term = 5
new_equation = 19 + constant_term
print(new_equation)

The output of the code is $24$. So, the equation becomes $-3w = 24$.

Solving for w

Now that we have isolated the variable $w$, we can solve for its value. To do this, we can divide both sides of the equation by the coefficient of $w$, which is $-3$.

# Solving for w
w_value = new_equation / -3
print(w_value)

The output of the code is $-8$. So, the value of $w$ is $-8$.

Conclusion

In this article, we have solved a linear equation of the form $-9w - 5 + 6w = 19$ to isolate the variable $w$. We have broken down the solution into manageable steps, making it easy to understand and follow along. By combining like terms and performing algebraic operations, we have isolated the variable $w$ and solved for its value.

Frequently Asked Questions

• Q: What is a linear equation? A: A linear equation is an equation in which the highest power of the variable(s) is 1.
• Q: How do I solve a linear equation? A: To solve a linear equation, you need to isolate the variable(s) on one side of the equation by combining like terms and performing algebraic operations.
• Q: What is the difference between a linear equation and a quadratic equation? A: A linear equation has the highest power of the variable(s) as 1, while a quadratic equation has the highest power of the variable(s) as 2.

Additional Resources

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

Final Answer

The final answer is: $\boxed{-8}$

Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In our previous article, we solved a linear equation of the form $-9w - 5 + 6w = 19$ to isolate the variable $w$. In this article, we will provide a Q&A guide to help students understand and solve linear equations.

Q&A Guide

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) on one side of the equation by combining like terms and performing algebraic operations. Here are the steps to follow:

1. Combine like terms on the left-hand side of the equation.
2. Add or subtract the same value to both sides of the equation to isolate the variable(s).
3. Multiply or divide both sides of the equation by the same value to isolate the variable(s).

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

A: A linear equation has the highest power of the variable(s) as 1, while a quadratic equation has the highest power of the variable(s) as 2. For example, the equation $x^2 + 4x + 4 = 0$ is a quadratic equation, while the equation $2x + 3 = 5$ is a linear equation.

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

A: To solve a linear equation with fractions, you need to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. For example, if the equation is $\frac{2}{3}x + 1 = \frac{5}{6}$, you can multiply both sides of the equation by 6 to eliminate the fractions.

Q: Can I use a calculator to solve a linear equation?

A: Yes, you can use a calculator to solve a linear equation. However, it's always a good idea to check your answer by plugging it back into the original equation to make sure it's true.

Q: What if I have a linear equation with multiple variables?

A: If you have a linear equation with multiple variables, you can solve it by using the same steps as before. However, you may need to use substitution or elimination methods to isolate the variables.

Q: Can I use algebraic properties to solve a linear equation?

A: Yes, you can use algebraic properties to solve a linear equation. For example, you can use the distributive property to expand expressions, or the commutative property to rearrange terms.

Examples

Example 1: Solving a Linear Equation with Fractions

Solve the equation $\frac{2}{3}x + 1 = \frac{5}{6}$.

# Solving a linear equation with fractions
from fractions import Fraction

# Define the equation
equation = Fraction(2, 3) * x + 1 == Fraction(5, 6)

# Multiply both sides of the equation by 6
new_equation = 4 * x + 6 == 5

# Subtract 6 from both sides of the equation
new_equation = 4 * x == -1

# Divide both sides of the equation by 4
x_value = -1 / 4

print(x_value)

Example 2: Solving a Linear Equation with Multiple Variables

Solve the equation $2x + 3y = 5$ and $x - 2y = -3$.

# Solving a linear equation with multiple variables
from sympy import symbols, Eq, solve

# Define the variables
x, y = symbols('x y')

# Define the equations
eq1 = Eq(2*x + 3*y, 5)
eq2 = Eq(x - 2*y, -3)

# Solve the system of equations
solution = solve((eq1, eq2), (x, y))

print(solution)

Conclusion

In this article, we have provided a Q&A guide to help students understand and solve linear equations. We have covered topics such as combining like terms, isolating variables, and using algebraic properties to solve linear equations. We have also provided examples of solving linear equations with fractions and multiple variables.

Frequently Asked Questions

• Q: What is a linear equation? A: A linear equation is an equation in which the highest power of the variable(s) is 1.
• Q: How do I solve a linear equation? A: To solve a linear equation, you need to isolate the variable(s) on one side of the equation by combining like terms and performing algebraic operations.
• Q: What is the difference between a linear equation and a quadratic equation? A: A linear equation has the highest power of the variable(s) as 1, while a quadratic equation has the highest power of the variable(s) as 2.

Additional Resources

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

Final Answer

The final answer is: $\boxed{-\frac{1}{4}}$