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Chapter 2: Problem 22

Solve each linear equation. $$ -12+8(x-5)=-4+3(5 x-2) $$

Short Answer

Expert verified
x = -6

Step by step solution

01

Distribute and Simplify

Distribute the numbers within each parenthesis to simplify the equation. This means multiplying 8 by each term inside the parenthesis and 3 by each term inside the other parenthesis. -12 + 8(x - 5) = -4 + 3(5x - 2) = -12 + 8x - 40 = -4 + 15x - 6.
02

Combine Like Terms

Now, combine like terms on each side of the equation. -12 - 40 + 8x = -4 - 6 + 15x -52 + 8x = -10 + 15x.
03

Solve for x

To solve for x, first isolate the variable on one side of the equation by subtracting 8x from both sides: -52 + 8x - 8x = -10 + 15x - 8x -52 = -10 + 7x Next, add 10 to both sides to isolate the term with x: -52 + 10 = 7x -42 = 7x Finally, divide by 7 to solve for x: x = -6.

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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 allows you to multiply a single term by each term inside a set of parentheses. This law is crucial for solving linear equations where you often need to eliminate parentheses. In our exercise, the equation is \[-12 + 8(x-5) = -4 + 3(5x-2)\]. Let's apply the distributive property step-by-step: For the left side: 8 multiplied by each term inside \((x-5)\) gives us \[8x - 40\]. So the left side becomes \[-12 + 8x - 40\]. For the right side: 3 multiplied by each term inside \((5x-2)\) results in \[15x - 6\]. So the right side becomes \[-4 + 15x - 6\]. Now the equation looks like this: \[-12 + 8x - 40 = -4 + 15x - 6\].
Combining Like Terms
After applying the distributive property, the next step is to simplify the equation further by combining like terms. These are terms that have the same variable or are constants. From our simplified equation \[-12 + 8x - 40 = -4 + 15x - 6\]: Combine like terms on each side of the equation:
  • On the left side \(-12 - 40 + 8x\) becomes \(-52 + 8x\).
  • On the right side \(-4 - 6 + 15x\) simplifies to \(-10 + 15x\).
We end up with the equation \[-52 + 8x = -10 + 15x\].
Isolating Variables
Next, it's time to isolate the variable term to one side of the equation. This makes it easier to solve for the variable. For our equation \[-52 + 8x = -10 + 15x\], we need to get all the x terms on one side and constants on the other side: Subtract 8x from both sides: \[-52 + 8x - 8x = -10 + 15x - 8x\]. This simplifies to \[-52 = -10 + 7x\]. Add 10 to both sides to get the constants on one side: \[-52 + 10 = 7x\]. Now, we have \[-42 = 7x\].
Solving for x
Finally, solving for x involves isolating the variable completely. From the equation \[-42 = 7x\], we need to get x by itself: Divide both sides by 7: \[\frac{-42}{7} = \frac{7x}{7}\]. This simplifies to \[x = -6\]. Therefore, our solution for x is \(-6\). Each of these steps is essential to solving linear equations, making them useful tools for handling a variety of algebraic problems.

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