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Chapter 2: Problem 33

Solve each equation, and check your solution. $$ 3 x+7=2 x+4 $$

Short Answer

Expert verified
x = -3

Step by step solution

01

- Isolate the variable term

Start by isolating the term containing the variable on one side of the equation. Subtract 2x from both sides: 3x + 7 - 2x = 2x + 4 - 2x.
02

- Simplify

Simplify the equation by combining like terms: (3x - 2x) + 7 = 4. This simplifies to: x + 7 = 4.
03

- Solve for x

Isolate x by subtracting 7 from both sides of the equation: x + 7 - 7 = 4 - 7. This simplifies to: x = -3.
04

- Check the solution

Substitute x = -3 back into the original equation to verify: 3(-3) + 7 = 2(-3) + 4. This simplifies to: -9 + 7 = -6 + 4. Simplify further: -2 = -2. Since both sides of the equation are equal, the solution is verified to be correct.

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Key Concepts

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

Isolating the Variable
Isolating the variable is crucial in solving linear equations. This means we need to get the variable, in this case, x, alone on one side of the equation. To achieve this, we perform operations that simplify the equation step by step.
In the given problem, we start with: \[ 3x + 7 = 2x + 4 \]First, we need to remove the term containing the variable from one side. Here, we subtract 2x from both sides: \[ 3x + 7 - 2x = 2x + 4 - 2x \]which simplifies to: \[ x + 7 = 4 \]Now, the variable x is isolated on the left side.
Combining Like Terms
Combining like terms helps simplify equations by merging terms that contain the same variable or are constants.
In our current equation: \[ x + 7 = 4 \]we see that we have isolated x, and now we have constants on both sides. This means we can combine these terms to further simplify. Subtract 7 from both sides:\[ x + 7 - 7 = 4 - 7 \]This results in:\[ x = -3 \]Now, we have fully simplified the equation to find the value of x.
Checking Solutions
It's important to check your solution to confirm it's correct. We do this by substituting the found value back into the original equation and verifying it holds true.
Let's substitute x = -3 back into the original equation: \[ 3(-3) + 7 = 2(-3) + 4 \]which simplifies to:\[ -9 + 7 = -6 + 4 \]Further simplifying, we get:\[ -2 = -2 \] Since both sides are equal, our solution is confirmed correct. Always perform this check to ensure no mistakes were made during the solving process.

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