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Chapter 5: Problem 308

In the following exercises, solve the equation. $$ -2.9 x=5.8 $$

Short Answer

Expert verified
x = -2

Step by step solution

01

Understand the equation

The given equation is \[ -2.9x = 5.8 \] To solve for \[ x \], you need to isolate \[ x \] on one side of the equation.
02

Divide both sides by -2.9

To isolate \[ x \], divide both sides of the equation by -2.9: \[ x = \frac{5.8}{-2.9} \]
03

Simplify the division

Solve the division on the right-hand side: \[ x = \frac{5.8}{-2.9} = -2 \]

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Key Concepts

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

Isolate the Variable
To solve a linear equation, your main goal is to isolate the variable. This means getting the variable on one side of the equation and everything else on the other side. In our example, the variable is \( x \).

To isolate \( x \), we have to undo any operation that is currently applied to \( x \). In the equation \(-2.9x = 5.8\), \( x \) is being multiplied by -2.9. So, we need to use the opposite of multiplication (which is division) to get rid of -2.9.

This is where the next concept, the Division Property of Equality, comes into play.
Division Property of Equality
The Division Property of Equality states that if you divide both sides of an equation by the same nonzero number, the two sides remain equal. This is a fundamental property of equations and is very useful in solving linear equations.

In our problem, we use this property to isolate \( x \). The equation is \(-2.9x = 5.8\) and we want to isolate \( x \) on one side. To do this, divide both sides by \(-2.9\):

\[ \frac{-2.9x}{-2.9} = \frac{5.8}{-2.9} \]

By dividing both sides by -2.9, you essentially cancel out the -2.9 on the left side, leaving \( x \) by itself:

\[ x = \frac{5.8}{-2.9} \]

Now \( x \) is isolated, and you can solve the fraction on the right-hand side.
Simplifying Fractions
Simplifying fractions is an important step that makes the expression easier to understand. In our equation, we have the fraction \( \frac{5.8}{-2.9} \).

To simplify this fraction, you perform the division indicated. In this case, 5.8 divided by -2.9 simplifies to -2:

\[ x = \frac{5.8}{-2.9} = -2 \]

It's always good practice to double-check your simplification. You can do this by performing the division again, even using a calculator to verify the result.

Understanding and simplifying fractions is a necessary skill not just for solving equations but for many areas of mathematics.

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