7x-7=3x-23. Can U Show The Steps
Solving algebraic equations can sometimes feel overwhelming, especially if you're new to the subject. However, with a clear step-by-step approach, solving equations like 7x - 7 = 3x - 23 becomes manageable and straightforward. In this comprehensive guide, we'll walk through the process of solving this equation, explaining each step in detail, so you can understand the reasoning behind each move. Whether you're a student preparing for exams or just brushing up on algebra, this article will help you master solving linear equations with confidence.
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Understanding the Equation
Before diving into the solution, it's essential to understand what the equation represents. The equation:
\[ 7x - 7 = 3x - 23 \]
is a linear equation in one variable, \( x \). Our goal is to find the value of \( x \) that makes both sides of the equation equal.
Key concepts:
- Linear equations involve variables raised only to the first power.
- Solving involves finding the value of the variable that satisfies the equation.
- Equivalent operations such as addition, subtraction, multiplication, and division can be performed on both sides of the equation to isolate the variable.
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Step-by-Step Solution Process
Let's now proceed through the steps to solve the equation:
\[ 7x - 7 = 3x - 23 \]
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Step 1: Simplify Both Sides of the Equation
In this case, the equation is already simplified since there are no like terms to combine on either side. However, always check for similar terms to combine.
Tip: Combining like terms reduces the complexity of the equation and makes solving easier.
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Step 2: Get All Variable Terms on One Side
To isolate \( x \), we want to gather all terms containing \( x \) on one side of the equation.
Method:
- Subtract \( 3x \) from both sides to move the \( 3x \) term to the left:
\[ 7x - 3x - 7 = 3x - 3x - 23 \]
which simplifies to:
\[ (7x - 3x) - 7 = 0 - 23 \]
Calculations:
\[ 4x - 7 = -23 \]
Result:
\[ 4x - 7 = -23 \]
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Step 3: Isolate the Variable Term
Next, eliminate the constant term on the side with the variable.
- Add 7 to both sides:
\[ 4x - 7 + 7 = -23 + 7 \]
which simplifies to:
\[ 4x = -16 \]
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Step 4: Solve for \( x \)
Now, divide both sides by the coefficient of \( x \), which is 4:
\[ \frac{4x}{4} = \frac{-16}{4} \]
which simplifies to:
\[ x = -4 \]
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Final Answer
The solution to the equation 7x - 7 = 3x - 23 is:
\[ \boxed{x = -4} \]
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Verification of the Solution
Always verify your solution by substituting \( x = -4 \) back into the original equation to ensure it satisfies both sides.
Original Equation:
\[ 7x - 7 = 3x - 23 \]
Substitute \( x = -4 \):
Left side:
\[ 7(-4) - 7 = -28 - 7 = -35 \]
Right side:
\[ 3(-4) - 23 = -12 - 23 = -35 \]
Since both sides equal \(-35\), the solution \( x = -4 \) is correct.
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Additional Tips for Solving Linear Equations
- Always perform the same operation on both sides to maintain equality.
- Simplify as much as possible at each step.
- Check your solution by substitution.
- Be cautious with negative signs and distribute multiplication carefully.
Common Mistakes to Avoid
- Forgetting to perform the operation on both sides (e.g., subtracting only from one side).
- Mixing up signs when moving terms across the equation.
- Not simplifying expressions fully before proceeding.
- Dividing by zero (which occurs if the coefficient of \( x \) becomes zero).
Practice Problems
To reinforce your understanding, try solving these similar equations:
- \( 5x + 3 = 2x + 9 \)
- \( 4x - 2 = 10 \)
- \( 3(x - 4) = 2x + 1 \)
- \( 6x + 7 = 2(3x + 4) \)
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Conclusion
Solving the equation 7x - 7 = 3x - 23 demonstrates fundamental algebraic techniques such as combining like terms, maintaining equality, and isolating the variable. By following the systematic steps outlined—gathering variables on one side, simplifying constants, and dividing by the coefficient—you can confidently solve similar linear equations. Remember to verify your solutions to ensure accuracy, and keep practicing with various equations to build your skills.
Mastery of solving equations is a crucial step in understanding more advanced algebra concepts, and with consistent practice, you'll become proficient at tackling these problems efficiently.
