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Chapter 2: Problem 43

Solve and check. Label any contradictions or identities. $$2(3+4 m)-6=48$$

Short Answer

Expert verified
m = 6. The solution is verified as correct.

Step by step solution

01

Distribute the 2

First, distribute the 2 to both terms inside the parentheses:2(3 + 4m) - 6 = 48This becomes:6 + 8m - 6 = 48
02

Combine like terms

Next, combine the constant terms on the left side of the equation:6 - 6 + 8m = 48This simplifies to:8m = 48
03

Solve for m

Divide both sides of the equation by 8 to solve for m:\(m = \frac{48}{8}\)\(m = 6\)
04

Check the solution

Substitute m = 6 back into the original equation to verify the solution:2(3 + 4(6)) - 6 = 48Calculate inside the parentheses:2(3 + 24) - 6 = 48Continue simplifying:2(27) - 6 = 4854 - 6 = 4848 = 48The solution is verified.

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Key Concepts

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

Distributive Property
The distributive property is a key concept in algebra that helps when you need to eliminate parentheses from an equation. It states that multiplying a number by a sum is the same as multiplying each addend by the number and then adding the products. In the given equation, the distributive property is used to simplify the expression:
  • Start with: \(2(3 + 4m) - 6 = 48\)
  • Distribute the 2 to both 3 and 4m: \(2 \times 3 + 2 \times 4m - 6 = 48\)
  • This gives us: \(6 + 8m - 6 = 48\)
By using the distributive property, we simplify the equation and prepare it for the next steps.
Combining Like Terms
After using the distributive property, combining like terms is the next crucial step. This involves simplifying expressions by adding or subtracting terms that have the same variable or are constants. In our equation:
  • The equation after distribution is: \(6 + 8m - 6 = 48\)
  • Combine the constants 6 and -6: \(6 - 6 + 8m = 48\)
  • This simplifies to: \(8m = 48\)
Combining like terms helps us to further reduce the equation to a simpler form, making it easier to solve for the variable.
Substitution Method
The substitution method often follows after simplifying an equation. This method is used to solve for the unknown variable by plugging the found values back into the original equation to check if they satisfy the equation. For our problem:
  • We solved \(8m = 48\) by dividing both sides by 8:
    \(m = \frac{48}{8}\)
    \(m = 6\)
  • Substitute \(m = 6\) back into the original equation: \(2(3 + 4(6)) - 6 = 48\)
This method helps in verifying that the solution derived is correct and satisfies the original equation.
Checking Solutions
Checking solutions ensures that the values found for the unknown variables are correct. This is a critical step in solving equations, as it confirms the accuracy of your solution. For our exercise:
  • Substituted \(m = 6\) back into the original equation: \(2(3 + 4(6)) - 6 = 48\)
Simplify inside the parentheses:
  • \(2(3 + 24) - 6 = 48\)
  • Which simplifies to: \(2(27) - 6 = 48\)
  • Results in: \(54 - 6 = 48\)
  • And finally confirms: \(48 = 48\)
Thus, \(m = 6\) is indeed the correct solution. Checking your answers prevents errors and reinforces the correctness of your solution.

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Most popular questions from this chapter

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