Solving a two-step equation with a fraction involves isolating the variable (the letter representing the unknown value) on one side of the equation. This process requires performing inverse operations to simplify the equation and find the value of the variable.
The steps to solve a two-step equation with a fraction are:
- Simplify any fractions in the equation.
- Undo the multiplication or division by multiplying or dividing both sides by the reciprocal of the coefficient of the variable.
- Combine like terms on each side of the equation.
- Solve for the variable by performing the remaining operation.
For example, to solve the equation:
(1/3)x + 2 = 5
- Multiply both sides by 3 to undo the multiplication by 1/3: x + 6 = 15
- Subtract 6 from both sides to isolate x: x = 9
1. Simplify
Simplifying fractions is a crucial step in solving two-step equations with fractions. Fractions represent parts of a whole, and simplifying them means expressing them in their simplest form, where the numerator and denominator have no common factors other than 1. This simplification process involves identifying and canceling out any common factors between the numerator and denominator, resulting in an equivalent fraction with the smallest possible numerator and denominator.
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Identifying Common Factors:
To simplify fractions, we first need to identify any common factors between the numerator and denominator. Common factors are numbers that divide both the numerator and denominator without leaving a remainder. For example, in the fraction 6/12, both 6 and 12 are divisible by 3, making 3 a common factor.
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Canceling Out Common Factors:
Once we have identified the common factors, we can cancel them out by dividing both the numerator and denominator by those factors. This process reduces the fraction to its simplest form. Continuing with the example above, we can cancel out the common factor 3 from both the numerator and denominator, resulting in the simplified fraction 2/4.
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Equivalent Fractions:
Simplifying a fraction does not change its value. The simplified fraction is an equivalent fraction, meaning it represents the same quantity as the original fraction. For instance, 2/4 is equivalent to 6/12, even though they have different numerators and denominators.
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Importance in Solving Equations:
Simplifying fractions is essential in solving two-step equations with fractions. It allows us to work with fractions in their simplest form, making the subsequent steps of solving the equation easier. By simplifying fractions, we can avoid unnecessary calculations and potential errors, leading to more accurate and efficient solutions.
In summary, simplifying fractions in two-step equations with fractions is a fundamental step that involves identifying and canceling out common factors to obtain equivalent fractions in their simplest form. This process ensures accurate and efficient equation solving.
2. Inverse Operations
Inverse operations play a critical role in solving two-step equations with fractions. When solving such equations, we often encounter multiplication or division operations involving fractions. To isolate the variable on one side of the equation, we need to undo these operations using inverse operations.
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Undoing Multiplication:
If a fraction is multiplied by a number, we can undo this multiplication by dividing both sides of the equation by that number. For example, if we have the equation (1/2)x = 5, we can undo the multiplication by dividing both sides by 1/2, which gives us x = 10.
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Undoing Division:
If a fraction is divided by a number, we can undo this division by multiplying both sides of the equation by that number. For example, if we have the equation x / (1/3) = 6, we can undo the division by multiplying both sides by 1/3, which gives us x = 2.
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Using Reciprocals:
The reciprocal of a number is the number that, when multiplied by the original number, gives us 1. When undoing multiplication or division by a fraction, we can use the reciprocal of that fraction. For example, the reciprocal of 1/2 is 2, and the reciprocal of (1/3) is 3.
By understanding and applying inverse operations, we can effectively solve two-step equations with fractions. This skill is essential for solving more complex equations and problems involving fractions.
3. Combine Like Terms
Combining like terms is a fundamental step in solving two-step equations with fractions. Like terms are terms that have the same variable and the same exponent. When we combine like terms, we add or subtract their coefficients while keeping the variable and exponent the same.
For example, in the equation 2x + 5 = 13, 2x and 5 are like terms because they both have the variable x and no exponent. We can combine them by adding their coefficients, which gives us 2x + 5 = 8.
Combining like terms is important because it simplifies the equation and makes it easier to solve. By combining like terms, we can reduce the number of terms in the equation and focus on the essential parts.
In the context of solving two-step equations with fractions, combining like terms allows us to isolate the variable on one side of the equation. This is because when we combine like terms, we can move all the terms with the variable to one side and all the constants to the other side.
For example, in the equation (1/2)x + 3 = 7, we can combine the like terms (1/2)x and 3 to get (1/2)x + 3 = 7. Then, we can isolate the variable by subtracting 3 from both sides, which gives us (1/2)x = 4.
Combining like terms is a crucial step in solving two-step equations with fractions because it simplifies the equation and allows us to isolate the variable. This skill is essential for solving more complex equations and problems involving fractions.
4. Solve for Variable
In the context of “How to Solve 2 Step Equation With Fraction”, the step “Solve for Variable: Perform the remaining operation to isolate the variable” is crucial for finding the value of the variable in the equation. After simplifying fractions, undoing multiplication or division, and combining like terms, the remaining operation is typically a simple arithmetic operation, such as addition or subtraction, that needs to be performed to isolate the variable on one side of the equation.
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Isolating the Variable:
The purpose of isolating the variable is to determine its value. By performing the remaining operation, we move all the terms containing the variable to one side of the equation and all the constant terms to the other side. This allows us to solve for the variable by dividing both sides of the equation by the coefficient of the variable.
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Solving for x:
In the context of two-step equations with fractions, the variable we are solving for is typically denoted by x. By performing the remaining operation and isolating the variable, we find the value of x that satisfies the equation.
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Example:
Consider the equation (1/2)x + 3 = 7. After simplifying the fraction and combining like terms, we get (1/2)x = 4. To solve for x, we perform the remaining operation of multiplication by 2 to both sides of the equation, which gives us x = 8. Therefore, the value of the variable x in this equation is 8.
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Importance:
Solving for the variable is the ultimate goal of solving two-step equations with fractions. It allows us to determine the unknown value that satisfies the equation and provides the solution to the problem.
In summary, the step “Solve for Variable: Perform the remaining operation to isolate the variable” is essential in “How to Solve 2 Step Equation With Fraction” because it enables us to find the value of the variable in the equation, which is the primary objective of solving the equation.
FAQs about “How to Solve 2-Step Equations with Fractions”
Question 1: What is the first step in solving a 2-step equation with a fraction?
Answer: The first step is to simplify any fractions in the equation.
Question 2: How do I undo multiplication or division when solving a 2-step equation with a fraction?
Answer: To undo multiplication, divide both sides of the equation by the coefficient of the variable. To undo division, multiply both sides by the coefficient of the variable.
Question 3: What is the purpose of combining like terms when solving a 2-step equation with a fraction?
Answer: Combining like terms simplifies the equation and makes it easier to isolate the variable.
Question 4: How do I isolate the variable when solving a 2-step equation with a fraction?
Answer: To isolate the variable, perform the remaining operation (addition or subtraction) to move all the terms containing the variable to one side of the equation and all the constant terms to the other side.
Question 5: What is the final step in solving a 2-step equation with a fraction?
Answer: The final step is to solve for the variable by performing the remaining operation (multiplication or division).
Question 6: Why is it important to be able to solve 2-step equations with fractions?
Answer: Solving 2-step equations with fractions is a fundamental skill in mathematics that is used in various applications, such as solving real-world problems and understanding algebraic concepts.
Summary: Solving 2-step equations with fractions involves simplifying fractions, undoing multiplication or division, combining like terms, isolating the variable, and solving for the variable. Understanding these steps is essential for solving these equations accurately and efficiently.
Tips for Solving 2-Step Equations with Fractions
Solving 2-step equations with fractions requires a systematic approach and attention to detail. Here are some tips to help you succeed:
Tip 1: Simplify Fractions
Before performing any operations, simplify all fractions in the equation to their simplest form. This will make the subsequent steps easier and reduce the risk of errors.
Tip 2: Understand Inverse Operations
When undoing multiplication or division involving fractions, use the concept of inverse operations. Multiply by the reciprocal of the coefficient to undo multiplication, and divide by the reciprocal to undo division.
Tip 3: Combine Like Terms
Combine terms with the same variable and exponent on each side of the equation. This will simplify the equation and make it easier to isolate the variable.
Tip 4: Isolate the Variable
To solve for the variable, isolate it on one side of the equation by performing the remaining operation (addition or subtraction). Move all terms containing the variable to one side and all constant terms to the other side.
Tip 5: Solve for the Variable
Once the variable is isolated, perform the final operation (multiplication or division) to find its value. This will give you the solution to the equation.
Tip 6: Check Your Answer
After solving the equation, substitute the value of the variable back into the original equation to verify if it satisfies the equation.
Summary:
By following these tips, you can develop a strong understanding of how to solve 2-step equations with fractions. Remember to simplify fractions, use inverse operations, combine like terms, isolate the variable, solve for the variable, and check your answer to ensure accuracy.
Conclusion
Solving two-step equations with fractions requires a systematic approach involving the simplification of fractions, understanding of inverse operations, combination of like terms, isolation of the variable, and solving for the variable. By following these steps and applying the tips discussed earlier, you can effectively solve these equations and expand your mathematical abilities.
The ability to solve two-step equations with fractions is a fundamental skill that serves as a building block for more complex algebraic concepts. It enables us to solve real-world problems, deepen our understanding of mathematical relationships, and develop critical thinking skills. By mastering this topic, you lay a solid foundation for your future mathematical endeavors.
