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Chapter 3: Problem 71

Use the properties of equality to help solve each equation. $$ -3.4 x=17 $$

Short Answer

Expert verified
The solution for \(x\) is \(-5\).

Step by step solution

01

Understand the Equation

The equation provided is \[-3.4x = 17\]. This is a linear equation where we need to solve for the variable \(x\). The equation tells us that \(-3.4\) times \(x\) equals 17.
02

Isolate the Variable

To solve for \(x\), we need to isolate \(x\) on one side of the equation. Since \(-3.4\) is multiplying \(x\), we can divide both sides of the equation by \(-3.4\) to solve for \(x\).The equation becomes:\[x = \frac{17}{-3.4}\]
03

Perform the Division

Now, we carry out the division to find the value of \(x\). Calculate \(\frac{17}{-3.4}\), which equals approximately \(-5\).So, \(x = -5\).

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Key Concepts

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

Properties of Equality
When solving linear equations, we often rely on the properties of equality. These properties are like golden rules that help keep equations balanced while we manipulate them. One of the most important properties is that if you add, subtract, multiply, or divide both sides of an equation by the same number, the two sides remain equal.

This means that whatever you do to one side of the equation, you must also do to the other.
  • Addition Property: If you add the same number to both sides of an equation, the equality is maintained.
  • Subtraction Property: Subtracting the same number from both sides also keeps the balance.
  • Multiplication Property: Multiplying both sides by the same nonzero number does not change the equality.
  • Division Property: You can divide both sides by the same nonzero number to solve for the variable.
Understanding and applying these properties ensures that we solve equations accurately without altering the true relationship between the sides of the equation.
Division in Algebra
Division plays a crucial role in solving algebraic equations, especially when a variable is multiplied by a number. When you encounter an equation like \(-3.4x = 17\), division helps us to isolate the variable.

To solve for \(x\), we need to "undo" the multiplication by dividing both sides by \(-3.4\). This use of division is actually relying on the Division Property of Equality.

It's like reversing multiplication:
  • You multiply during the equation setup to form it, and you divide to break it back down.
  • This lets us clear the coefficient (the number multiplying our variable) from the equation.
In our example, dividing both sides by \(-3.4\) gives us \(x = \frac{17}{-3.4}\). This step is essential to reach the solution.
Isolating Variables
Isolating variables is a key part of solving equations. The goal is to get the variable by itself on one side of the equation so that we can easily identify its value.

In our example, we started with the equation \(-3.4x = 17\). To isolate \(x\), we used division. By dividing every term in the equation by \(-3.4\), we shifted \(x\) to stand alone, resulting in the equation: \(x = \frac{17}{-3.4}\). This tells us the value of \(x\).

When isolating variables:
  • Always perform the same operation on both sides to maintain balance.
  • If the variable is in a more complex expression, simplify systematically.
  • Focus on undoing operations surrounding the variable to "free" it.
Once the variable is isolated, you can determine its value, just as we did here, finding that \(x\) is approximately \(-5\).

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