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Chapter 2: Problem 35

Solve and check. Label any contradictions or identities. $$4 x-6=6 x$$

Short Answer

Expert verified
The solution is \(x = -3\). The solution verifies correctly without contradictions or identities.

Step by step solution

01

- Combine like terms

Subtract \(4x\) from both sides of the equation to combine the \(x\) terms: \[ 4x - 6 - 4x = 6x - 4x \] This simplifies to \[ -6 = 2x \]
02

- Solve for x

Divide both sides of the equation by 2 to solve for \(x\): \[ \frac{-6}{2} = \frac{2x}{2} \] This simplifies to \[ x = -3 \]
03

- Check the solution

Substitute \(x = -3\) back into the original equation to verify the solution: \[ 4(-3) - 6 = 6(-3) \] Calculate both sides: \[ -12 - 6 = -18 \] \[ -18 = -18 \] Both sides are equal, so the solution is verified.
04

- Identify contradictions or identities

Since the equation simplifies to \(-18 = -18\) after substituting \(x = -3\), this does not present a contradiction or identity beyond the solution already derived. Thus, there are no contradictions or special identities here.

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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 step in solving linear equations. It simplifies the equation by grouping similar terms together. For instance, in the given equation, we subtracted \(4x\) from both sides.

This process is essential as it reduces the number of terms, making it easier to solve the equation. Like terms are terms that have the same variable raised to the same power.

For example:
  • \(3x\) and \(4x\) are like terms because they both have the variable \(x\).
  • \(-7\) and \(-5\) are like terms because they are both constants.
Combining them correctly helps us to focus on solving one variable step by step.
Solving for x
Once like terms are combined, the next goal is to isolate the variable \(x\). The equation from the previous step, \(-6 = 2x\), requires us to divide both sides by 2 to solve for \(x\).

The process involves:
  • Dividing \(-6\) by 2
  • Dividing \(2x\) by 2
This simplifies to \(x = -3\).

This step ensures we find the exact value of \(x\) that satisfies the equation. Always divide or multiply the entire equation by the same number to maintain balance.
Checking Solutions
Once we find the solution, it鈥檚 essential to check if it鈥檚 correct. Substitute \(x = -3\) back into the original equation: \(4(-3) - 6 = 6(-3)\).

Calculating both sides, we get:
  • Left side: \(-12 - 6 = -18\)
  • Right side: \(-18\)
Since both sides are equal, the solution \(-3\) is verified. This step confirms our solution is accurate and ensures no mistakes were made during the solving process.

It鈥檚 a good practice to always verify your solution to confirm its correctness.
Equation Verification
The final step is to check for any contradictions or identities. Here, since both sides of the equation are equal, \(-18 = -18\), we've verified the solution does not lead to any contradictions.Identifying contradictions or identities means:
  • A contradiction occurs if substituting the solution back makes an illogical statement (e.g., \(5 = 7\)).
  • An identity occurs if all values of the variable make the equation true (e.g., \(x = x\)).
In this case, the solution \(-3\) is valid without any contradictions or identities beyond the simplifications we鈥檝e already performed.

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Most popular questions from this chapter

Solve and check. Label any contradictions or identities. $$3(y+4)=3(y-1)$$

Solve using the multiplication principle. Don't forget to check! $$ 84=7 n $$

Use an inequality and the five-step process to solve each problem. Rhetoric Advertising is a directmail company. It determines that for a particular campaign, it can use any envelope with a fixed width of \(3 \frac{1}{2}\) in. and an area of at least \(17 \frac{1}{2}\) in \(^{2} .\) Determine (in terms of an inequality) those lengths that will satisfy the company constraints.

Insert the symbol \(<,>, \leq\) or \(\geq\) to make each pair of inequalities equivalent. $$ -7 t \geq 56 ; t \square-8 $$

Use an inequality and the five-step process to solve each problem. The formula \(R=-0.0065 t+4.3259\) can be used to predict the world record, in minutes, for the 1 -mi run \(t\) years after \(1900 .\) Determine (in terms of an inequality) those years for which the world record will be less than \(3.6 \mathrm{min}\)
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