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Chapter 2: Problem 67

Solve each equation, and check your solution. $$ 10(-2 x+1)=-19(x+1) $$

Short Answer

Expert verified
x = 29

Step by step solution

01

- Distribute the constants

Distribute the constants 10 and -19 to the terms inside the parentheses. This will give us: \[ 10(-2x) + 10(1) = -19(x) + -19(1) \Rightarrow -20x + 10 = -19x - 19 \]
02

- Combine like terms

Move all the terms involving x to one side of the equation and the constants to the other side. Add 19x to both sides and subtract 10 from both sides: \[ -20x + 19x + 10 - 10 = -19x + 19x - 19 - 10 \Rightarrow -x = -29 \]
03

- Solve for x

Solve for x by dividing both sides by -1: \[ x = 29 \]
04

- Check your solution

Substitute the value of x back into the original equation to verify the solution: \[ 10(-2(29) + 1) = -19(29 + 1) \Rightarrow 10(-58 + 1) = -19(30) \Rightarrow 10(-57) = -570 \Rightarrow -570 = -570 \] The 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 simplify expressions. It allows you to multiply a single term by terms inside parentheses.
By distributing, we break down expressions and make them easier to work with. For example, in the equation:
10(-2x + 1) = -19(x + 1)
To apply the distributive property, multiply each term inside the parentheses by the factor outside.
Here鈥檚 how it works in our example:
  • 10 * -2x = -20x
  • 10 * 1 = 10
  • -19 * x = -19x
  • -19 * 1 = -19
Putting it all together, the equation simplifies to: -20x + 10 = -19x - 19 This is much easier to handle.
Combining Like Terms
After distributing, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power.
In our simplified equation: -20x + 10 = -19x - 19
Let's look at the x-terms and constants more closely:
  • Move the x-terms to one side by adding 19x to both sides: -20x + 19x = -19x + 19x - 19 + 10
  • Combine them: -1x = -29
By combining these like terms, we get closer to isolating the variable.
Verifying Solutions
Verification is the final step to ensure our solution is correct. We substitute the found value back into the original equation:
  • Our solution for x was 29. Substitute x = 29 back into the original equation to check: 10(-2(29) + 1) = -19(29 + 1)
  • Calculate each side: 10(-58 + 1) = -19(30) 10(-57) = -570-570 = -570
Since both sides of the equation are equal, our solution is verified. Verifying solutions ensures that no mistakes were made during solving.

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