Solve For $d$.$\[ \begin{array}{l} -9d - 4d - 9d + 9 = -13 \\ d = \square \end{array} \\]

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 type of linear equation, where the variable is isolated on one side of the equation. We will use a step-by-step approach to solve the equation and provide a clear understanding of the process.

The Equation

The given equation is:

$$-9d - 4d - 9d + 9 = -13$$

Our goal is to isolate the variable $d$ on one side of the equation.

Step 1: Combine Like Terms

The first step in solving the equation is to combine like terms. In this case, we have three terms with the variable $d$, and one constant term.

$$-9d - 4d - 9d + 9 = -13$$

We can combine the three terms with the variable $d$ by adding their coefficients.

$$-22d + 9 = -13$$

Step 2: Isolate the Variable

Now that we have combined like terms, we can isolate the variable $d$ by moving the constant term to the other side of the equation.

$$-22d = -13 - 9$$

We can simplify the right-hand side of the equation by subtracting 9 from -13.

$$-22d = -22$$

Step 3: Solve for $d$

Now that we have isolated the variable $d$, we can solve for its value by dividing both sides of the equation by -22.

$$d = \frac{-22}{-22}$$

We can simplify the right-hand side of the equation by canceling out the -22 in the numerator and denominator.

$$d = 1$$

Conclusion

In this article, we have solved a linear equation by combining like terms and isolating the variable $d$. We have used a step-by-step approach to provide a clear understanding of the process. By following these steps, students can master the skill of solving linear equations and apply it to a wide range of mathematical problems.

Tips and Tricks

• When combining like terms, make sure to add or subtract the coefficients of the terms with the same variable.
• When isolating the variable, move the constant term to the other side of the equation by adding or subtracting the same value to both sides.
• When solving for the variable, divide both sides of the equation by the coefficient of the variable.

Real-World Applications

Linear equations have numerous real-world applications, including:

• Physics: Linear equations are used to describe the motion of objects under constant acceleration.
• Engineering: Linear equations are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Linear equations are used to model economic systems and make predictions about future trends.

Common Mistakes

• Failing to combine like terms correctly.
• Failing to isolate the variable correctly.
• Failing to solve for the variable correctly.

Practice Problems

1. Solve the equation: $2x + 5x - 3x = 7$
2. Solve the equation: $-4y - 2y + 3y = -11$
3. Solve the equation: $x + 2x - 3x = -5$

Conclusion

Introduction

In our previous article, we discussed the step-by-step process of solving linear equations. However, we understand that sometimes, students may have questions or doubts about the process. In this article, we will address some of the most frequently asked questions about solving linear equations.

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable is 1. In other words, it is an equation that can be written in the form ax + b = c, where a, b, and c are constants.

Q: How do I know if an equation is linear?

A: To determine if an equation is linear, look for the highest power of the variable. If it is 1, then the equation is linear. For example, the equation 2x + 3 = 5 is linear because the highest power of x is 1.

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. For example, the equation x^2 + 2x + 1 = 0 is a quadratic equation because the highest power of x is 2.

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

A: To solve a linear equation with fractions, follow the same steps as solving a linear equation with integers. However, you may need to multiply both sides of the equation by the least common multiple (LCM) of the denominators to eliminate the fractions.

Q: What is the order of operations when solving a linear equation?

A: When solving a linear equation, follow the order of operations (PEMDAS):

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 isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, follow these steps:

1. Add or subtract the same value to both sides of the equation to eliminate any constants on the same side as the variable.
2. Multiply or divide both sides of the equation by the same value to eliminate any coefficients on the same side as the variable.

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

A: A linear equation is a single equation with one variable, while a system of linear equations is a set of two or more linear equations with the same variables.

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

A: To solve a system of linear equations, follow these steps:

1. Write the equations in the form ax + by = c.
2. Use the method of substitution or elimination to solve for one variable.
3. Substitute the value of the variable into one of the original equations to solve for the other variable.

Conclusion

In conclusion, solving linear equations is a crucial skill for students to master. By following the step-by-step process outlined in this article, students can solve linear equations with confidence and apply it to a wide range of mathematical problems. If you have any further questions or doubts, feel free to ask!

Practice Problems

1. Solve the equation: 2x + 3 = 5
2. Solve the equation: x - 2 = 3
3. Solve the system of equations: x + y = 4, 2x - y = 2

Common Mistakes

• Failing to combine like terms correctly.
• Failing to isolate the variable correctly.
• Failing to solve for the variable correctly.

Tips and Tricks

• When solving a linear equation, always follow the order of operations (PEMDAS).
• When isolating the variable, make sure to add or subtract the same value to both sides of the equation.
• When solving a system of linear equations, use the method of substitution or elimination to solve for one variable.