In mathematics, an equation is a statement that asserts the equality of two expressions. Equations with variables on both sides are equations in which the variable appears on both the left and right-hand sides of the equation. For example, the equation $$3x – 5 = 2x + 7$$ has the variable \(x\) on both sides of the equation.
Checking equations with variables on both sides is important because it allows us to determine whether a given value of the variable makes the equation true. To check an equation with variables on both sides, we can use the following steps:
- Subtract the constant term from both sides of the equation.
- Combine like terms on both sides of the equation.
- Factor out the coefficient of the variable on both sides of the equation.
- Divide both sides of the equation by the coefficient of the variable.
- Simplify both sides of the equation.
If the final result is a true statement, then the original equation is true for the given value of the variable. If the final result is a false statement, then the original equation is not true for the given value of the variable.
1. Isolate the variable
Isolating the variable is a crucial step in checking equations with variables on both sides. Without isolating the variable, it would be impossible to solve for its value. This is because the variable would be mixed in with other terms on both sides of the equation, making it difficult to determine its exact value.
By isolating the variable, we can simplify the equation and make it easier to solve. For example, consider the equation $$3x – 5 = 2x + 7$$. To solve for x, we would first isolate the variable on one side of the equation. We can do this by adding 5 to both sides of the equation: $$3x – 5 + 5 = 2x + 7 + 5$$ $$3x = 2x + 12$$
Now that the variable is isolated on one side of the equation, we can solve for its value. We can do this by subtracting 2x from both sides of the equation: $$3x – 2x = 2x + 12 – 2x$$ $$x = 12$$
Therefore, the solution to the equation $$3x – 5 = 2x + 7$$ is $$x = 12$$.
Isolating the variable is a fundamental step in checking equations with variables on both sides. By isolating the variable, we can simplify the equation and make it easier to solve.
2. Combine like terms
Combining like terms is a crucial step in checking equations with variables on both sides. Without combining like terms, it would be difficult to simplify the equation and solve for the variable.
For example, consider the equation $$3x – 5 = 2x + 7$$. To solve for x, we would first isolate the variable on one side of the equation. We can do this by adding 5 to both sides of the equation:
$$3x – 5 + 5 = 2x + 7 + 5$$$$3x = 2x + 12$$
Now that the variable is isolated on one side of the equation, we can combine like terms on both sides. In this case, the only like terms are the two terms with \(x\):
$$3x – 2x = 12$$
We can combine these terms by subtracting \(2x\) from both sides of the equation:
$$x = 12$$
Therefore, the solution to the equation $$3x – 5 = 2x + 7$$ is \(x = 12\).
Combining like terms is a fundamental step in checking equations with variables on both sides. By combining like terms, we can simplify the equation and make it easier to solve.
3. Simplify
Simplifying an equation by dividing both sides by the coefficient of the variable is a crucial step in checking equations with variables on both sides. Without this step, it would be difficult to solve for the variable and determine whether the equation is true or false.
- Isolating the variable: Dividing both sides of the equation by the coefficient of the variable allows us to isolate the variable on one side of the equation. This makes it easier to see the value of the variable and to check whether the equation is true or false.
- Solving for the variable: Once the variable is isolated, we can solve for the variable by dividing both sides of the equation by the coefficient of the variable. This will give us the value of the variable.
- Checking the solution: After we have solved for the variable, we can check the solution by substituting it back into the original equation. If the equation is true, then the solution is correct.
Simplifying an equation by dividing both sides by the coefficient of the variable is a fundamental step in checking equations with variables on both sides. By following this step, we can ensure that our solutions are correct and that we have a complete understanding of the equation.
4. Check the solution
Checking the solution is an important step in how to check equations with variables on both sides. Without this step, we cannot be sure that our solution is correct. Substituting the value of the variable back into the original equation allows us to verify that the equation is true. If the equation is true, then we know that our solution is correct.
For example, consider the equation $$3x – 5 = 2x + 7$$. We can solve this equation by isolating the variable on one side of the equation:
$$3x – 5 + 5 = 2x + 7 + 5$$$$3x = 2x + 12$$$$x = 12$$
Once we have solved for \(x\), we can check our solution by substituting \(x = 12\) back into the original equation:
$$3(12) – 5 = 2(12) + 7$$$$36 – 5 = 24 + 7$$$$31 = 31$$
Since the equation is true, we know that our solution is correct.
Checking the solution is a fundamental step in how to check equations with variables on both sides. By following this step, we can ensure that our solutions are correct and that we have a complete understanding of the equation.
FAQs on How to Check Equations with Variables on Both Sides
Checking equations with variables on both sides is a fundamental mathematical skill. It allows us to determine whether a given value of the variable makes the equation true.
Here are six frequently asked questions (FAQs) on how to check equations with variables on both sides:
Question 1: What is the first step in checking equations with variables on both sides?
Answer: The first step is to isolate the variable on one side of the equation.
Question 2: How do I isolate the variable?
Answer: To isolate the variable, add or subtract the same value from both sides of the equation.
Question 3: What is the next step after isolating the variable?
Answer: The next step is to combine like terms on both sides of the equation.
Question 4: What is the purpose of combining like terms?
Answer: Combining like terms simplifies the equation and makes it easier to solve for the variable.
Question 5: How do I solve for the variable?
Answer: To solve for the variable, divide both sides of the equation by the coefficient of the variable.
Question 6: What is the final step in checking equations with variables on both sides?
Answer: The final step is to check the solution by substituting it back into the original equation.
By following these steps, you can check equations with variables on both sides and ensure that your solutions are correct.
For more information on how to check equations with variables on both sides, please refer to the following resources:
- Khan Academy: Checking Equations with Variables on Both Sides
- Math is Fun: Checking Equations
- Purple Math: Solving Equations with Variables on Both Sides
Tips on How to Check Equations with Variables on Both Sides
Understanding how to check equations with variables on both sides is a valuable mathematical skill. Utilize these essential tips to enhance your problem-solving abilities:
Tip 1: Isolate the Variable
Isolate the variable by performing the same operations (addition or subtraction) on both sides of the equation. Focus on moving the variable to one side and constants to the other.
Tip 2: Combine Like Terms
Identify and combine terms with the same variable and exponent on both sides of the equation. This simplifies the equation, making it easier to solve.
Tip 3: Solve for the Variable
Once the variable is isolated, solve for it by dividing both sides of the equation by the coefficient of the variable. This will yield the value of the variable.
Tip 4: Check Your Solution
Substitute the obtained value of the variable back into the original equation. If the equation holds true, your solution is accurate.
Tip 5: Practice Regularly
Consistency is key. Engage in regular practice to strengthen your understanding and proficiency in checking equations with variables on both sides.
By implementing these tips, you can effectively check equations with variables on both sides, ensuring accurate solutions and a deeper comprehension of algebraic equations.
Remember, consistent practice and a methodical approach are essential for mastering this mathematical skill.
The Significance of Checking Equations with Variables on Both Sides
Checking equations with variables on both sides is a fundamental mathematical skill that enables us to verify the validity of algebraic expressions. This process involves isolating the variable, combining like terms, solving for the variable, and checking the solution. By systematically following these steps, we can ensure accurate results and enhance our understanding of algebraic equations.
The ability to check equations with variables on both sides is essential for solving more complex algebraic problems and applications. It provides a solid foundation for advanced mathematical concepts and empowers us to tackle real-world scenarios where algebraic equations are used to model and solve problems.