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Chapter 5: Problem 7

Solve and check. $$2 a-5=7$$

Short Answer

Expert verified
The solution for the equation is \(a = 6\). Checking by substituting the value of 'a' back in the original equation confirms that the solution is correct.

Step by step solution

01

Isolate the term with the variable

The goal is to isolate 'a' on one side of the equation. First, try to eliminate other quantities from the left side where 'a' is located. Add 5 to both sides of the equation \(2a-5+5 = 7+5\). The 'minus 5' and 'plus 5' on the left side will cancel each other out.
02

Simplify the Equation

After performing the operation from Step 1, the equation simplifies to \(2a = 12\). So, now 'a' is multiplied by 2.
03

Solve for 'a'

To isolate 'a' completely, divide both sides of the equation by 2: \(2a/2 = 12/2\).
04

Check the solution

To confirm if the solution for 'a' is correct, substitute 6 (our obtained solution) back into the original equation and verify if both sides are equal: Substitute \(a = 6\) in \(2a-5=7\) and check the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Equation Solving
Solving an equation is like finding the missing piece in a puzzle. You're looking for the unknown or variable in the equation that makes both sides equal. To solve an equation, follow a systematic approach where you simplify terms and strategically perform operations.
This process ensures you're keeping the equation balanced.

Steps in equation solving generally include:
  • Identifying the variable you need to solve for (in this case, 'a').
  • Performing operations to get the variable alone on one side of the equation.
  • Simplifying the equation after each operation.
  • Checking your work by substituting the solution back into the original equation.
Each step helps inch you closer to solving the equation. The goal is to simplify and isolate the variable to see what value makes the equation true.
Variable Isolation
When solving linear equations, variable isolation is a crucial step. It involves moving terms around to get the variable by itself on one side of the equation.
This lets you clearly see what the variable equals.

In the given problem, **2a - 5 = 7**, our target is to isolate 'a':
  • First, eliminate any constants or numbers associated with the variable. Here, you add 5 to both sides, cancelling out the '鈭�5'. This operation gives 2a = 12
  • Next, focus on the variable's coefficient. Divide both sides by 2 to solve for 'a'. This gives a = 6.
Steps like adding, subtracting, multiplying, or dividing both sides of the equation ensure balance. Balancing means any change you make to one side is done to the other. This is key in maintaining the equation's equality.
Substitution Method
The substitution method is a way to verify your solution by plugging the value you found back into the original equation. For the equation **2a - 5 = 7**, after solving the equation and finding 'a' to be 6, you substitute 'a' with 6 to check if both sides of the equation are equal.

Here's how it works:
  • Substitute the value a = 6 back into the original equation. This gives 2(6) - 5 = 7.
  • Compute the left-hand side: 2 times 6 equals 12, minus 5 equals 7.
  • Since both sides equal 7, your solution is verified.
This step acts as a safety check, confirming the accuracy of your solution. It ensures that no mistakes were made during the calculations. Using substitution is important as it reassures you that the derived solution truly satisfies the original equation.

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