Equations are like sentences that show that two things are equal. They usually have letters (called variables) that stand for unknown numbers, along with numbers (called constants) and operators that show what kind of math to do (like + for addition, - for subtraction, × for multiplication, and ÷ for division).
Steps to Solve Equations with Variables on Both Sides
When solving equations that have variables on both sides, the main aim is to move all variable terms to one side of the equation and the constant terms to the other side.
Below are the following steps can be followed:
Step 1: Simplify Both Sides: Expand any brackets and simplify the terms on both sides if needed.
Step 2: Move Variables to the One Side: Use addition or subtraction to get all variables on one side.
Step 3: Move the Constants to the Other Side: Use addition or subtraction to move all constants to the other side.
Step 4: Isolate Variable: Solve for a variable using the inverse operations (e.g., divide or multiply).
Step 5: Check Your Solution: Substitute your answer back into the original equation to verify.
Let's discuss an example for better understanding.
Solved Examples on Solving Equations with Variables on Both Sides
Example 1: Solve for x: 5x - 3 = 2x + 9
Solution:
Step 1: Move the variable to one side
Subtract 2x from both the sides: 5x - 2x - 3 = 9
Simplifies to: 3x - 3 = 9Step 2: Move the constants to other side
Add 3 to both the sides: 3x = 12Step 3: Isolate variable
Divide both the side by 3: x = 4Step 4: Check your solution
Substitute x = 4 back into original equation:
5(4) - 3 = 2(4) + 9
⇒ 20 - 3 = 8 + 9
⇒ 17 = 17Since the both sides are equal, solution x = 4 is correct.
Example 2: Solve equation 6x + 4 = 3x + 10
Solution:
Subtract 3x from the both sides to get all variables on one side:
6x - 3x + 4 = 10Simplifies to: 3x + 4 = 10
Subtract 4 from both to isolate term with x: 3x = 6
Divide both sides by 3: x = 2Answer: x = 2
Solved Questions on Solving Equation with Variables on Both Side
Question 1: Solve the equation: 6x + 4 = 3x + 10
Solution:
Subtract 3x from both sides: 6x - 3x + 4 = 10
Simplifies to: 3x + 4 = 10
Subtract 4 from both sides: 3x = 6
Divide both sides by 3: x = 2.Answer: x = 2
Question 2: Solve the equation: 8x - 5 = 3x + 10
Solution:
Subtract 3x from both sides: 8x - 3x -5 =10
Amplifies to 5x - 5 = 10
Add 5 to both sides: 5x = 15
Divide by 5: x = 3Answer: x =3
Question 3: Solve the equation: 4(x+2) = 3x + 14
Solution:
Expand the left side: 4x + 8 = 3x + 14
Subtract 3x from both the sides: 4x - 3x + 8 = 14
Simplifies to: x + 8 = 14
Subtract 8 from both sides: x = 6Answer: x = 6
Question 4: Solve the equation: 2(3x - 1) = 4x + 8
Solution:
Expand left side: 6x - 2 = 4x + 8
Subtract 4x from both sides: 6x - 4x - 2 = 8
Simplifies to: 2x - 2 = 8
Add 2 to both the sides: 2x = 10
Divide by 2: x = 5Answer: x = 5
Question 5: Solve Equation: 7x - 3 = 5x + 11
Solution:
Move variable terms to one side by subtracting 5x from both sides: 7x - 5x - 3 = 11
Simplifies to: 2x - 3 = 11
Move constant term to other side by adding 3 to both sides: 2x = 14
Divide both sides by 2: x = 7Answer: x = 7
Practice Questions on Solving Equations with Variable on Both Sides
You can download free PDF for worksheet on Solving Equations with Variable on Both Sides from below:
Download Free Worksheet on Solving Equations with Variable on Both Sides |
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Conclusion
In conclusion, solving equations with variables on both sides is an essential skill in algebra that allows us to find unknown values efficiently. By simplifying both sides, moving variables, and using inverse operations, we can isolate the variable and determine the solution.
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