Solve For { X $} . . . { 5 - 4 + 7x + 1 = \} ${ \text{ex} + \square }$
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 involves isolating the variable x. We will use a step-by-step approach to solve the equation, and provide explanations and examples to help illustrate the process.
What is a Linear Equation?
A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. Linear equations can be written in the form ax + b = c, where a, b, and c are constants. In the equation we will be solving, 5 - 4 + 7x + 1 = 0, the highest power of x is 1, making it a linear equation.
The Equation to be Solved
The equation we will be solving is:
5 - 4 + 7x + 1 = 0
Step 1: Simplify the Equation
The first step in solving the equation is to simplify it by combining like terms. In this case, we can combine the constants 5, -4, and 1.
5 - 4 + 1 = 2
So, the simplified equation is:
2 + 7x = 0
Step 2: Isolate the Variable x
The next step is to isolate the variable x by getting all the terms with x on one side of the equation. We can do this by subtracting 2 from both sides of the equation.
2 + 7x - 2 = 0 - 2
This simplifies to:
7x = -2
Step 3: Solve for x
The final step is to solve for x by dividing both sides of the equation by 7.
7x / 7 = -2 / 7
This simplifies to:
x = -2/7
Conclusion
Solving linear equations is an essential skill for students to master. By following the steps outlined in this article, we have successfully solved the equation 5 - 4 + 7x + 1 = 0. The solution to the equation is x = -2/7. We hope this article has provided a clear and concise guide to solving linear equations.
Example Problems
Here are a few example problems to help reinforce the concepts learned in this article.
Example 1
Solve the equation 3x + 2 = 5.
Step 1: Simplify the Equation
3x + 2 = 5
Subtract 2 from both sides:
3x = 5 - 2
3x = 3
Step 2: Isolate the Variable x
Divide both sides by 3:
x = 3 / 3
x = 1
Example 2
Solve the equation 2x - 3 = 7.
Step 1: Simplify the Equation
2x - 3 = 7
Add 3 to both sides:
2x = 7 + 3
2x = 10
Step 2: Isolate the Variable x
Divide both sides by 2:
x = 10 / 2
x = 5
Example 3
Solve the equation x + 4 = 9.
Step 1: Simplify the Equation
x + 4 = 9
Subtract 4 from both sides:
x = 9 - 4
x = 5
Conclusion
Solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, we have successfully solved three example problems. We hope this article has provided a clear and concise guide to solving linear equations.
Tips and Tricks
Here are a few tips and tricks to help you solve linear equations:
- Always simplify the equation by combining like terms.
- Isolate the variable x by getting all the terms with x on one side of the equation.
- Use inverse operations to solve for x.
- Check your solution by plugging it back into the original equation.
Q: What is a linear equation?
A: A linear equation is an equation in which the highest power of the variable (in this case, x) is 1. Linear equations can be written in the form ax + b = c, where a, b, and c are constants.
Q: How do I simplify a linear equation?
A: To simplify a linear equation, combine like terms by adding or subtracting the coefficients of the same variable. For example, in the equation 2x + 3x = 5, combine the like terms 2x and 3x to get 5x = 5.
Q: How do I isolate the variable x?
A: To isolate the variable x, get all the terms with x on one side of the equation by using inverse operations. For example, in the equation 2x + 3 = 5, subtract 3 from both sides to get 2x = 2, and then divide both sides by 2 to get x = 1.
Q: What is an inverse operation?
A: An inverse operation is an operation that "reverses" another operation. For example, addition and subtraction are inverse operations, as are multiplication and division. In the equation 2x + 3 = 5, the inverse operation of addition is subtraction, and the inverse operation of multiplication is division.
Q: How do I check my solution?
A: To check your solution, plug it back into the original equation and see if it is true. For example, if you solve the equation 2x + 3 = 5 and get x = 1, plug x = 1 back into the original equation to get 2(1) + 3 = 5, which is true.
Q: What if I have a fraction as a coefficient?
A: If you have a fraction as a coefficient, you can multiply both sides of the equation by the denominator of the fraction to eliminate the fraction. For example, in the equation 2/3x = 4, multiply both sides by 3 to get 2x = 12.
Q: What if I have a negative coefficient?
A: If you have a negative coefficient, you can multiply both sides of the equation by -1 to eliminate the negative sign. For example, in the equation -2x = 4, multiply both sides by -1 to get 2x = -4.
Q: Can I solve a linear equation with multiple variables?
A: Yes, you can solve a linear equation with multiple variables by using the same steps as before. For example, in the equation 2x + 3y = 5, isolate one variable by getting all the terms with that variable on one side of the equation, and then solve for the other variable.
Q: What if I have a linear equation with a variable on both sides?
A: If you have a linear equation with a variable on both sides, you can add or subtract the variable on both sides to eliminate the variable. For example, in the equation x + 2x = 5, add x to both sides to get 3x = 5.
Conclusion
Solving linear equations is a crucial skill for students to master. By following the steps outlined in this article, you can solve linear equations with ease. Remember to simplify the equation, isolate the variable, and check your solution. With practice, you will become a master of solving linear equations.
Additional Resources
- Khan Academy: Solving Linear Equations
- Mathway: Solving Linear Equations
- IXL: Solving Linear Equations
By using these resources, you can practice solving linear equations and improve your skills.