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Chapter 2: Problem 12

Solve and check. Label any contradictions or identities. $$6 x-3=15$$

Short Answer

Expert verified
x = 3

Step by step solution

01

- Add 3 to Both Sides

To isolate the term with the variable, add 3 to both sides of the equation: \[6x - 3 + 3 = 15 + 3\]Simplifying this, we get:\[6x = 18\]
02

- Divide Both Sides by 6

Next, divide both sides of the equation by 6 to solve for x: \[\frac{6x}{6} = \frac{18}{6}\]This simplifies to:\[x = 3\]
03

- Check the Solution

To ensure the solution is correct, substitute x = 3 back into the original equation and verify the left-hand side equals the right-hand side:\[6(3) - 3 = 15\]Solve this to see if it holds true:\[18 - 3 = 15\]Since both sides are equal, x = 3 is correct.
04

- Label Any Contradictions or Identities

Since solving the equation resulted in a specific value for x and there were no contradictions or identities, we conclude that the equation is neither contradictory nor an identity.

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Key Concepts

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

addition property of equality
In solving linear equations, the addition property of equality is a fundamental step. It states that you can add the same number to both sides of an equation without changing the equality. This is crucial for isolating the variable in an equation.
In our example, the equation is: 6x - 3 = 15.
To start solving for x, we need to eliminate the -3 term on the left side. By adding 3 to both sides of the equation, we use the addition property of equality:
6x - 3 + 3 = 15 + 3.
Simplifying this, we get:6x = 18.
This step helps to set up the equation so that the variable term, 6x, is isolated, making further steps straightforward.
division property of equality
After applying the addition property, the next step is to use the division property of equality. This property allows you to divide both sides of the equation by the same non-zero number without changing the equality. In our example, the equation is:6x = 18.
To solve for x, we need to eliminate the coefficient 6 that is multiplying x by dividing both sides of the equation by 6:\[\frac{6x}{6} = \frac{18}{6}\].
Simplifying this, we get:x = 3.
This step isolates x and gives us the solution. By ensuring that we divide both sides evenly, we maintain the balance of the equation and solve for the variable accurately.
checking solutions
Once we have found a solution, it's essential to check that it satisfies the original equation. This step ensures that our solution is accurate. To check our solution x=3, we substitute x back into the original equation:6x - 3 = 15.
Substituting the value of x:6(3) - 3 = 15.
Solving this, we find:18 - 3 = 15.
Since both sides of the equation are equal, our solution x=3 is correct.
Always remember to check your solutions to verify their correctness. If the left-hand side of the equation equals the right-hand side after substitution, the solution is validated. Otherwise, re-evaluate your steps for any errors.

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