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Chapter 2: Problem 70

For exercises 65-86, (a) solve. (b) check. $$ 4(3 x-8)=2(6 x-1)+x-30 $$

Short Answer

Expert verified
x = 0

Step by step solution

01

- Distribute and Expand

Distribute the constants inside the parenthesis: 4(3x - 8) = 2(6x - 1) + x - 30Expands to:12x - 32 = 12x - 2 + x - 30
02

- Combine Like Terms

Combine like terms on the right side: 12x - 32 = 12x + x - 2 - 30Simplifies to:12x - 32 = 13x - 32
03

- Isolate the Variable

Subtract 12x from both sides: 12x - 32 - 12x = 13x - 32 - 12xSimplifies to:-32 = x - 32
04

- Solve for x

Add 32 to both sides to solve for x: -32 + 32 = x - 32 + 32Simplifies to:x = 0
05

- Check the Solution

Substitute x = 0 back into the original equation and verify both sides are equal:4(3(0)-8)=2(6(0)-1)+0-30Simplifies to:4(-8) = 2(-1) - 30-32 = -2 - 30 -32 = -32The solution x = 0 is correct.

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Key Concepts

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

distribution in algebra
Distribution in algebra involves multiplying each term inside a parenthesis by the factor outside it.
In the exercise, we distribute 4 and 2 like this:
  • 4(3x - 8) = 4 * 3x + 4 * (-8)
  • 2(6x - 1) = 2 * 6x + 2 * (-1)This results in:
    • 12x - 32
    • 12 x - 2
      • always apply this to simplify expressions before solving equations.
combining like terms
Combining like terms means adding or subtracting terms with the same variable. For instance, in the solution:
12x - 32 = 12x - 2 + x - 30
we combine the x terms and the constant terms, leading to:
  • 12x - 32 = 12x + x - 2 - 30
Then, it simplifies to:
  • 12x - 32 = 13x - 32
Combining like terms makes the equation easier to work with by reducing the number of terms.
isolating variables
Isolating the variable means getting the variable (x) by itself on one side of the equation. Start by eliminating any identical terms on both sides:
12x - 32 = 13x - 32
Subtract 12x from both sides:
12x - 32 - 12x = 13x - 32 - 12x
This simplifies to:
  • -32 = x - 32
Now, get x alone by adding 32 to both sides:
  • -32 + 32 = x - 32 + 32
    • - simplifies to:
    • x = 0
    The variable x is now isolated.
checking solutions
Checking solutions ensures that your solution is correct by plugging the variable back into the original equation:
4(3(0) - 8) = 2(6(0) - 1) + 0 - 30Simplify both sides:
  • -32 = -2 - 30
Both sides equal -32, so the solution x = 0 is verified correct. Always check your solutions to confirm their accuracy!

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