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Chapter 3: Problem 22

solve the given equation. If the equation is always true or has no solution, indicate this. $$x+5=5-x$$

Short Answer

Expert verified
The solution is \ x = 4 \.

Step by step solution

01

Identify the given equation

The given equation is \( -9x = -36 \). This is a simple linear equation with one variable, x.
02

Isolate the variable

To isolate the variable x, divide both sides of the equation by -9. This will help to get rid of the coefficient of x on the left-hand side. \( \frac{-9x}{-9} = \frac{-36}{-9} \).
03

Simplify

Simplify both sides of the equation: \[ x = \frac{-36}{-9} \]. This simplifies to: \[ x = 4 \].
04

Verify the solution

Substitute the value of x back into the original equation to verify: \(-9(4) = -36\). \(-36 = -36\) is true, which confirms that \ x = 4 \ is correct.

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Key Concepts

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

linear equations
Linear equations are equations where the highest power of the variable is one. They are fundamental in algebra and appear in the form: ax + b = c. In our example, the equation is -9x = -36. Here, -9 is the coefficient of the variable x, and -36 is a constant. Linear equations have unique or infinite solutions depending on their form. Understanding the structure of linear equations helps in manipulating and solving them efficiently.
isolation of variables
Isolation of variables means rearranging the equation so that the variable is by itself on one side. This focuses our efforts on finding the variable's value. In the example -9x = -36, we want to get x alone. To do this, we divide both sides of the equation by -9. -9x 梅 -9 = -36 梅 -9 This simplification isolates x on the left-hand side, leading us to the next step in solving the equation. Isolating variables is crucial because it allows us to find specific values quickly and accurately.
equation verification
After finding the value of the variable, it's important to verify our solution. Verification checks if our solution satisfies the original equation. For x = 4, we substitute it back into the original equation -9x = -36: -9(4) = -36 -36 = -36 Since both sides of the equation are equal, our solution is verified. This step ensures accuracy and confirms that we have solved the equation correctly. Verification acts as a final check in mathematical problem-solving.
simplification
Simplification involves reducing expressions to their simplest form. It's essential for making equations easier to understand and solve. In the given example, we simplified -36 梅 -9 to get 4: x = 4.Simplification helps in avoiding complex calculations and mistakes. It makes both sides of the equation easier to compare and work with, which is critical especially in more complex equations. Simplifying as you go helps in maintaining clarity and accuracy in your work. It's a key skill in mathematics that applies to solving different types of algebraic problems.

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Most popular questions from this chapter

Set up an inequality and solve it. Be sure to clearly label what the variable represents. The medium side of a triangle is \(2 \mathrm{cm}\) longer than the shortest side, and the longest side is twice as long as the shortest side. If the perimeter of the triangle is to be at least \(30 \mathrm{cm}\) and no more than \(50 \mathrm{cm},\) what is the range of values for the shortest side?

Solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. One of the various models that is proposed for the proper weight \(W\) (in pounds) of a man \(h\) inches tall is \(W=5.7 h-228\) (a) According to this model, what is the proper weight for a person who is 6 feet tall? (b) According to this model, how tall is a person whose proper weight is 200 pounds?

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Solve each of the problems algebraically. That is, set up an equation and solve it. Be sure to clearly label what the variable represents. Round your answer to the nearest tenth where necessary. Anthropologists often extrapolate the appearance of a human being from the parts of a skeleton they uncover. The following mathematical model relates the height \(h\) (in centimeters) of the individual to the length \(f\) (in centimeters) of the femur (thigh bone): $$h=3.68 f+21.44$$ (a) Use this model to find the height of an individual whose femur measures \(35 \mathrm{cm}\) (b) Use this model to find the length of a femur corresponding to an individual who is \(180 \mathrm{cm}\) tall.

Set up an inequality and solve it. Be sure to clearly label what the variable represents. If 9 less than 5 times a number is less than twice the number, how small must the number be?
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