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Chapter 2: Problem 51

Solve each equation, and check your solution. $$ 3 x+7-2 x=0 $$

Short Answer

Expert verified
The solution to the equation is \(x = -7\). It checks out when substituted back into the original equation.

Step by step solution

01

Combine like terms

Start by combining the like terms on the left side of the equation: i.e., combine the terms with the variable x: the equation is: \[3x - 2x + 7 = 0 \]to get:\[x + 7 = 0\]
02

Isolate the variable

Subtract 7 from both sides to isolate the variable x: equation now is:\[ x + 7 - 7 = 0 - 7 \]simplifies to:\[ x = -7 \]
03

Check the solution

Substitute the solution \(x = -7\) into the original equation \[3x + 7 - 2x = 0 \] and check if both sides are equal: \[ 3(-7) + 7 - 2(-7) = 0 \]evaluate to:\[ -21 + 7 + 14 = 0\]which simplifies to:\[0 = 0\]. Hence, the solution is verified.

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Key Concepts

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

Combining Like Terms
To solve linear equations efficiently, combining like terms is a crucial step. Like terms are terms that have the same variables raised to the same power. For example, in the equation \[3x + 7 - 2x = 0\] we have two terms involving \(x\): \(3x\) and \(-2x\). These are 'like' because they both have the variable \(x\). Combining these terms makes the equation simpler: \[3x - 2x + 7 = 0\] Now, you just need to perform the arithmetic operation, which leaves you with: \[x + 7 = 0\] By reducing the equation in this way, you make it easier to isolate the variable and solve for it.
Isolating the Variable
Once you have simplified the equation, the next step is isolating the variable. The goal is to get the variable by itself on one side of the equation. From our simplified equation \(x + 7 = 0\), you need to 'move' the constant term to the other side. You can do this by subtracting 7 from both sides: \[x + 7 - 7 = 0 - 7\] This simplifies to our solution: \[x = -7\] By isolating the variable, you have now found that \(x = -7\).
Checking Solutions in Algebra
It's always a good idea to check your solutions to ensure they are correct. To do this, substitute the solution back into the original equation and see if both sides of the equation balance. Our original equation was: \[3x + 7 - 2x = 0\] Substitute \(x = -7\) into the equation: \[3(-7) + 7 - 2(-7) = 0\] Now, evaluate each term: \[-21 + 7 + 14 = 0\] Combine these values to see if the result is 0: \[0 = 0\] As both sides of the equation are equal, the solution is correctly verified. Always make sure to check your solutions to confirm their accuracy.

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