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Chapter 2: Problem 55

Solve each equation, and check your solution. $$ 6 x+5+7 x+3=12 x+4 $$

Short Answer

Expert verified
x=-4

Step by step solution

01

Combine like terms on the left side

Combine the terms involving x: 6x and 7x, and the constant terms: 5 and 3. Thus, the equation becomes: 13x + 8 = 12x + 4
02

Move variables to one side

Subtract 12x from both sides to isolate the variable terms on one side: 13x - 12x + 8 = 12x - 12x + 4 This simplifies to: x + 8 = 4
03

Isolate the variable

Subtract 8 from both sides to isolate x: x + 8 - 8 = 4 - 8 This simplifies to: x = -4
04

Verify the solution

Substitute x = -4 back into the original equation to ensure it is correct: 6(-4) + 5 + 7(-4) + 3 = 12(-4) + 4 Simplify each term: -24 + 5 - 28 + 3 = -48 + 4 Simplify further: -44 = -44 Therefore, the solution x = -4 is verified as correct.

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Key Concepts

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

Combining Like Terms
Combining like terms is a crucial first step when solving linear equations. This process involves simplifying expressions by adding or subtracting terms that have the same variables or are constants.
For example, in the equation given:
\ 6x + 5 + 7x + 3 = 12x + 4 \ we notice both variable terms (\(6x\) and \(7x\)) and constant terms (5 and 3).
Combining them results in:
\ 13x + 8 = 12x + 4 \ Now, the equation is simpler to solve. Always look out for like terms to streamline the solving process.
Isolating Variables
Isolating the variable is the heart of solving linear equations. This step aims to get the variable on one side of the equation while constants are moved to the other.
Using our simplified equation: \ 13x + 8 = 12x + 4 \ We subtract \(12x\) from both sides:
\ 13x - 12x + 8 = 12x - 12x + 4 \ which simplifies to:
\ x + 8 = 4 \ The next part is straightforward; subtract the constant term (8) from both sides:
\ x + 8 - 8 = 4 - 8 \ resulting in: \ x = -4 \ By isolating the variable, you can easily determine its value.
Verifying Solutions
Verifying your solution is always important to ensure accuracy. To verify, substitute the found solution back into the original equation. Let's substitute \(x = -4\): \ 6(-4) + 5 + 7(-4) + 3 = 12(-4) + 4 \ Simplify each term: \(-24 + 5 - 28 + 3 = -48 + 4\).
Combining the terms results in: \ -44 = -44 \ Since both sides of the equation are equal, the solution is verified as correct.
Always double-check your solution by substituting it back into the initial equation to detect any errors you might have made during solving.

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