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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 both sides of the equation

First, distribute the constants on both sides of the equation. On the left side, distribute the 10, and on the right side, distribute the -19: \[ 10(-2x + 1) = -19(x + 1) \]\[ 10 \times -2x + 10 \times 1 = -19 \times x + -19 \times 1 \]This simplifies to: \[ -20x + 10 = -19x - 19 \]
02

- Move variable terms to one side

Next, move the variable terms to one side of the equation by adding 20x to both sides: \[ -20x + 10 + 20x = -19x - 19 + 20x \]This simplifies to: \[ 10 = x - 19 \]
03

- Solve for x

Now, solve for x by adding 19 to both sides of the equation: \[ 10 + 19 = x - 19 + 19 \]This simplifies to: \[ 29 = x \]
04

- Check the solution

Finally, check the solution by substituting x back into the original equation: \[ 10(-2(29) + 1) = -19(29 + 1) \]Simplify each side: \[ 10(-58 + 1) = -19(30) \]\[ 10 \times -57 = -570 \]\[ -570 = -570 \]The solution confirms that our value for x is correct.

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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 crucial concept in algebra. It allows you to simplify expressions by distributing a multiplier outside the parentheses to each term inside the parentheses.
In our exercise, the original equation is: \[ 10(-2x + 1) = -19(x + 1) \]
By applying the distributive property, you multiply each term inside the parentheses by the constant outside. So, it becomes:
\[ 10 \times -2x + 10 \times 1 = -19 \times x + -19 \times 1 \]
This simplifies the equation to:
\[ -20x + 10 = -19x - 19 \]
The distributive property makes the equation easier to manipulate further.
algebraic manipulation
Algebraic manipulation involves using various algebraic techniques to simplify and solve equations.
In our problem, the next step after distributing is to move the variable terms to one side.
We had:
\[ -20x + 10 = -19x - 19 \]
To isolate the variable x, add 20x to both sides:
\[ -20x + 10 + 20x = -19x - 19 + 20x \]
This gives us:
\[ 10 = x - 19 \]
Now, to find x, add 19 to both sides:
\[ 10 + 19 = x - 19 + 19 \]
Which simplifies to:
\[ 29 = x \]
This step-by-step manipulation helps us solve for the unknown variable.
verifying solutions
Verifying solutions is an essential step to ensure the accuracy of your answer.
To check our solution, we substitute \( x = 29 \) back into the original equation to see if both sides are equal:
Original equation: \[ 10(-2(29) + 1) = -19(29 + 1) \]
Simplify both sides:
\[ 10(-58 + 1) = -19(30) \]
\[ 10(-57) = -570 \]
\[ -570 = -570 \]
Both sides equal the same value, confirming our solution is correct.
Always double-check your work to avoid errors and solidify your understanding of the process.

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