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Chapter 2: Problem 51

Solve and check. Label any contradictions or identities. $$7+3 x-6=3 x+5-x$$

Short Answer

Expert verified
The solution is x = 4.

Step by step solution

01

Simplify Both Sides

Start by simplifying both sides of the equation. The left side is already simplified: 7 + 3x - 6 you can simplify this to 1 + 3x. On the right side, simplify 3x + 5 - x it becomes 2x + 5.
02

Set the Equation Equal

Set the simplified left side equal to the simplified right side: 1 + 3x = 2x + 5.
03

Isolate the Variable

To isolate the variable, first subtract 2x from both sides: 1 + 3x - 2x = 2x + 5 - 2x which simplifies to 1 + x = 5. Then subtract 1 from both sides: 1 + x - 1 = 5 - 1 which gives x = 4.
04

Check the Solution

Substitute x = 4 back into the original equation to verify: 7 + 3(4) - 6 = 3(4) + 5 - 4 check if both sides are equal: 7 + 12 - 6 = 12 + 5 - 4 13 = 13 Since both sides are equal, x = 4 is the correct solution.

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Key Concepts

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

Solving Linear Equations
Linear equations are equations of the first order. This means that each term is either a constant or a product of a constant and a single variable. In this exercise, our goal is to find the value of the variable that makes the equation true. For example, consider the equation given: \[7 + 3x - 6 = 3x + 5 - x\]. First, we transform and simplify both sides to make it easier to solve for the variable.
Simplifying Expressions
Simplifying expressions involves combining like terms to reduce the equation to its simplest form. On the left side of the given equation, \[7 + 3x - 6\], you can combine the constants: \[7 - 6\], resulting in \[1 + 3x\]. On the right side, \[3x + 5 - x\], you combine the coefficients of \(x\): \[3x - x\], which simplifies to \[2x + 5\]. Now our equation is reduced to \[1 + 3x = 2x + 5\].
Isolating the Variable
Isolating the variable means getting the variable on one side of the equation. First, subtract \(2x\) from both sides to get the equation \[1 + 3x - 2x = 2x + 5 - 2x\], which simplifies to \[1 + x = 5\]. Then, subtract 1 from both sides to find \[x\]: \[1 + x - 1 = 5 - 1\], simplifying to \[x = 4\]. This process leaves us with a straightforward value for \(x\).
Checking Solutions
After finding the value of the variable, checking the solution ensures its correctness. Substitute \(x = 4\) back into the original equation: \[7 + 3(4) - 6 = 3(4) + 5 - 4\]. Perform the arithmetic to see if both sides are equal: \[7 + 12 - 6 = 12 + 5 - 4\], which simplifies to \[13 = 13\]. Since both sides are equal, \(x = 4\) is indeed the correct solution.

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