Solving Linear Equations with Variable on both Sides Last Updated : 30 Jan, 2025 Comments Improve Suggest changes Like Article Like Report Equations consist of two main components: variables and numbers. Understanding the relationship between these components and how to manipulate them is essential for solving equations.Variable: A variable is a symbol (often a letter like x, y, or z) that represents an unknown or changing quantity.Number: A number is a constant value in an equation.Methods for Solving Equations with Variables on Both SidesWhen a variable appears on both sides of an equation, the following principles apply to simplify and solve the equation:Adding/Subtracting Terms: You can add or subtract the same number (or expression) to both sides of the equation without altering its equality.Multiplying/Dividing Terms: You can multiply or divide both sides of the equation by the same non-zero number without changing its equality.These operations are called reverse operations, and they help you isolate the variable to solve the equation.Example: Solve 14 - 2x = 5x for the value of x.Step 1: First, we need separate variables on one side and numbers on the other side by applying some basic operations.Add 2x on both the sides14 - 2x + 2x = 5x + 2x(Similarly, we can subtract a term with a variable from both sides of the equation)Step 2: Perform operations to convert the coefficient of the variable to 1.Equation: 14 = 7xDivide 7 on both the sidesx = 2Example: Solve 64 + 2x = 10x + 8 for the value of xStep 1: Subtract 2x from both sides:64 + 2x - 2x = 10x - 2x + 864 = 8x + 8Step 2: Subtract 8 from both the sides:64 - 8 = 8x + 8 - 856 = 8x Step 3: Divide 8 on both the sidesx = 7Note: In every problem of this kind it is always recommended separating the numbers and variables on either side of the equation by applying the reverse operations.Sample Problems on Linear Equations with variables on both SidesExample 1. Solve for x: 35x - 45 = 25Solution: Add 45 to both the sides 35x - 45 + 45 = 25 + 45 35x = 70Divide 35 on both the sides x = 2 Example 2. Solve for x: 22 - 32x = 33 + xSolution: Add 32x on both the sides 22 - 32x + 32x = 33 + x + 32x 22 = 33 + 33x Subtract 33 from both the sides 22 - 33 = 33 + 33x -33 -11 = 33x Divide 11 on both the sides-1 = 3x Divide 3 on both the sides x = -1/3Example 3. Solve for x: 23x + 4 = 104 + 3xSolution: Subtract 4 from both the sides 23x + 4 - 4 = 104 + 3x - 4 23x = 100 + 3x Subtract 3x from both the sides 23x - 3x = 100 + 3x - 3x 20x = 100 Divide 20 from both the sides x = 5Example 4. Solve for x: 45x + 21 = 15x + 141Solution:Subtract 21 from both the sides 45x + 21 - 21 = 15x + 141 - 21 45x = 15x + 120 Subtract 15x from both the sides 45x - 15x = 15x + 120 - 15x 30x = 120 Divide 30 on both the sides x = 4Example 5. Solve for x: 28x + 33 = 108 + 3xSolution:Subtract 3x from both the sides 28x + 33 -3x = 108 + 3x - 3x 25x + 33 = 108 Subtract 33 from both the sides 25x + 33 - 33 = 108 - 33 25x = 75 Divide 25 on both the sidesx = 3Example 6. Solve for x: 8x + 3x = 34 + 2 + 2xSolution: Simplify: 11x = 36 + 2x Get the variable on one side: 11x - 2x = 36 + 2x - 2x 9x = 36 Solve using inverse operations: x = 4Check Whether: 8(4) + 3(4) = 34 + 2 + 2(4)? Yes! Example 7. Solve for y: 33y - 32 = 19 - 18ySolution: The equation is already simplified. Get the variable on one side using inverse operations 33y - 32 = 19 - 18y 51y - 32 = 19 51y = 19 + 32 51y = 51y = 1Check: 33y - 32 = 19 - 18y? Yes! Solve Linear Equations with Variable on both Sides WorksheetProblem 1: Solve for x: 3x + 5 = 2x + 8Problem 2: Solve for y: 4y − 2 = 2y + 6.Problem 3: Solve for x: 5(x − 2) = 2x + 7Problem 4: Solve for z: 7z + 3 = 5z + 9Problem 5: Solve for x: 6x − 4 = 2x + 12Problem 6: Solve for a: 2(a + 3) = 4 + 3aProblem 7: Solve for b: 8 − 3b = 2b + 1Problem 8: Solve for m: 3m − 4 = 5(m + 2)Answer Keyx = 3y = 4x = 17/3z = 3x = 4a = 2b = 7/5m = −7 Comment More infoAdvertise with us Next Article Reducing Equations to Simpler Form | Class 8 Maths G gauthamd Follow Improve Article Tags : Mathematics School Learning Class 8 Linear Equations Maths-Class-8 +1 More Similar Reads CBSE Class 8th Maths Notes CBSE Class 8th Maths Notes cover all chapters from the updated NCERT textbooks, including topics such as Rational Numbers, Algebraic Expressions, Practical Geometry, and more. Class 8 is an essential time for students as subjects become harder to cope with. At GeeksforGeeks, we provide easy-to-under 15+ min read Chapter 1: Rational Numbers Rational NumbersA rational number is a type of real number expressed as p/q, where q ≠0. Any fraction with a non-zero denominator qualifies as a rational number. Examples include 1/2, 1/5, 3/4, and so forth. Additionally, the number 0 is considered a rational number as it can be represented in various forms such a 9 min read Natural Numbers | Definition, Examples & PropertiesNatural numbers are the numbers that start from 1 and end at infinity. In other words, natural numbers are counting numbers and they do not include 0 or any negative or fractional numbers.Here, we will discuss the definition of natural numbers, the types and properties of natural numbers, as well as 11 min read Whole Numbers - Definition, Properties and ExamplesWhole numbers are the set of natural numbers (1, 2, 3, 4, 5, ...) plus zero. They do not include negative numbers, fractions, or decimals. 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Discount is offered by the business owner to easily and quickly sell their product or services. Giving discounts increases the sales of the business and helps the business retain its customer. Discount is always 9 min read Sales Tax, Value Added Tax, and Goods and Services Tax - Comparing Quantities | Class 8 MathsTax is a mandatory fee levied by the government to collect revenue for public works providing the best facilities and infrastructure.The first known Tax system was in Ancient Egypt around 3000–2800 BC, in First Dynasty of Egypt. The first form of taxation was corvée and tithe. In India, The Tax was 5 min read Simple InterestSimple Interest (SI) is a method of calculating the interest charged or earned on a principal amount over a fixed period. It is calculated based solely on the principal amount, which remains unchanged throughout the calculation.Simple Interest is widely used across industries such as banking, financ 9 min read Compound Interest | Class 8 MathsCompound Interest: Compounding is a process of re-investing the earnings in your principal to get an exponential return as the next growth is on a bigger principal, following this process of adding earnings to the principal. In this passage of time, the principal will grow exponentially and produce 9 min read Compound InterestCompound Interest is the interest that is calculated against a loan or deposit amount in which interest is calculated for the principal as well as the previous interest earned. Compound interest is used in the banking and finance sectors and is also useful in other sectors. 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