Solve For \[$d\$\]:$\[ 5d + 5 = 2d - 4 \\]Options:A. \[$ D = \frac{7}{8} \$\] B. \[$ D = -3 \$\] C. \[$ D = -\frac{7}{9} \$\] D. \[$ D = 3 \$\]
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, which is a first-degree equation in one variable. We will use the given equation 5d + 5 = 2d - 4 as an example to demonstrate the step-by-step process of solving linear equations.
Understanding the Equation
Before we start solving the equation, let's understand what it represents. The equation 5d + 5 = 2d - 4 is a linear equation in one variable, which is 'd'. The equation states that the product of 5 and 'd' plus 5 is equal to the product of 2 and 'd' minus 4.
Step 1: Isolate the Variable
The first step in solving a linear equation is to isolate the variable. In this case, we need to isolate 'd'. To do this, we will use the distributive property to simplify the equation.
Distributive Property
The distributive property states that for any numbers a, b, and c:
a(b + c) = ab + ac
We can use this property to simplify the equation by distributing the numbers outside the parentheses to the terms inside.
Simplifying the Equation
Using the distributive property, we can rewrite the equation as:
5d + 5 = 2d - 4
becomes
5d + 5 - 5 = 2d - 4 - 5
This simplifies to:
5d = 2d - 9
Subtracting 2d from Both Sides
To isolate 'd', we need to get all the terms with 'd' on one side of the equation. We can do this by subtracting 2d from both sides of the equation.
5d - 2d = 2d - 9 - 2d
This simplifies to:
3d = -9
Dividing Both Sides by 3
Now that we have isolated 'd', we can solve for it by dividing both sides of the equation by 3.
3d / 3 = -9 / 3
This simplifies to:
d = -3
Conclusion
In this article, we solved a linear equation using the distributive property and algebraic manipulation. We isolated the variable 'd' and solved for it by dividing both sides of the equation by 3. The solution to the equation is d = -3.
Options
Now that we have solved the equation, let's compare our solution to the options provided.
- A. d = 7/8
- B. d = -3
- C. d = -7/9
- D. d = 3
Our solution, d = -3, matches option B.
Discussion
Solving linear equations is an essential skill for students to master. It requires a deep understanding of algebraic manipulation and the distributive property. In this article, we demonstrated the step-by-step process of solving a linear equation using these concepts. We also compared our solution to the options provided and found that our solution matches option B.
Practice Problems
If you want to practice solving linear equations, try the following problems:
- Solve the equation 2x + 5 = 3x - 2
- Solve the equation x - 3 = 2x + 1
- Solve the equation 4y - 2 = 2y + 5
Remember to use the distributive property and algebraic manipulation to isolate the variable and solve for it.
Conclusion
Introduction
In our previous article, we discussed the step-by-step process of solving linear equations using the distributive property and algebraic manipulation. In this article, we will provide a Q&A guide to help you better understand the concepts and solve linear equations with confidence.
Q: What is a linear equation?
A: A linear equation is a type of equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable.
Q: What is the distributive property?
A: The distributive property is a mathematical concept that states that for any numbers a, b, and c:
a(b + c) = ab + ac
This property allows us to simplify equations by distributing the numbers outside the parentheses to the terms inside.
Q: How do I isolate the variable in a linear equation?
A: To isolate the variable, you need to get all the terms with the variable on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
Q: What is the difference between a linear equation and a quadratic equation?
A: A linear equation is a type of equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable. A quadratic equation, on the other hand, is a type of equation that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable.
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 whole numbers. 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 least common multiple (LCM)?
A: The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. For example, the LCM of 2 and 3 is 6, because 6 is the smallest number that is a multiple of both 2 and 3.
Q: How do I solve a linear equation with decimals?
A: To solve a linear equation with decimals, you need to follow the same steps as solving a linear equation with whole numbers. However, you may need to multiply both sides of the equation by a power of 10 to eliminate the decimals.
Q: What is the difference between a linear equation and a system of linear equations?
A: A linear equation is a type of equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable. A system of linear equations, on the other hand, is a set of two or more linear equations that are solved simultaneously.
Q: How do I solve a system of linear equations?
A: To solve a system of linear equations, you need to follow the same steps as solving a linear equation. However, you may need to use substitution or elimination methods to solve the system.
Conclusion
Solving linear equations is a crucial skill for students to master. In this article, we provided a Q&A guide to help you better understand the concepts and solve linear equations with confidence. We hope this guide has been helpful in your journey to mastering linear equations.
Practice Problems
If you want to practice solving linear equations, try the following problems:
- Solve the equation 2x + 5 = 3x - 2
- Solve the equation x - 3 = 2x + 1
- Solve the equation 4y - 2 = 2y + 5
Remember to use the distributive property and algebraic manipulation to isolate the variable and solve for it.
Additional Resources
If you want to learn more about solving linear equations, we recommend the following resources:
- Khan Academy: Linear Equations
- Mathway: Linear Equations
- Wolfram Alpha: Linear Equations
We hope this guide has been helpful in your journey to mastering linear equations. Happy solving!