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Chapter 3: Problem 25

Solve each equation. $$-7 x+3=-7$$

Short Answer

Expert verified
The solution is \(x = \frac{10}{7}\).

Step by step solution

01

Isolate Variable

The equation given is \(-7x + 3 = -7\). The first step is to isolate the variable term \(-7x\) on one side by eliminating the constant on its side. Subtract 3 from both sides of the equation for an equivalent equation: \[-7x + 3 - 3 = -7 - 3\] This simplifies to: \[-7x = -10\].
02

Solve for the Variable

Now that we have the equation \(-7x = -10\), the next step is to solve for \(x\). We need to divide both sides by the coefficient of \(x\), which is -7, to isolate \(x\). \[x = \frac{-10}{-7}\].
03

Simplify the Solution

Simplify the fraction to find the value of \(x\). Since dividing a negative number by a negative number results in a positive number, the solution is: \[x = \frac{10}{7}\].

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Key Concepts

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

Isolating Variables
When solving a linear equation like \(-7x + 3 = -7\), the first major step is isolating the variable. Imagine you are trying to figure out the value of \(x\) by itself.
To accomplish this, it is crucial to "move" other numbers to the opposite side of the equation. This leaves \(x\) alone on one side, making it easier to find its value. Take the equation: \(-7x + 3 = -7\). To isolate \(-7x\), subtract 3 from both sides:
  • On the left, you have: \(-7x + 3 - 3 = -7x\)
  • On the right, you get: \(-7 - 3 = -10\)
As a result, the equation becomes \(-7x = -10\). We have successfully isolated the variable term.
Equation Simplification
Simplification is an important step when dealing with linear equations. It helps by reducing the equation to its simplest form.
Once you've isolated the variable term, the equation \(-7x = -10\) needs to be further simplified to solve for \(x\). You should aim to have \(x\) by itself. Therefore, you need to get rid of the coefficient attached to \(x\).
In this case, you divide both sides of the equation by -7. This operation is applied to both sides, keeping the equation balanced:
  • \(\frac{-7x}{-7} = \frac{-10}{-7}\)
Here, the left side simplifies to \(x\). Simple division allows us to handle the complex parts, making the solution accessible.
Rational Numbers
Rational numbers play a key role when solving and simplifying equations. A rational number is simply a number that can be expressed as a fraction, where both the numerator and the denominator are integers.
In our problem, once we solved for \(x\), we found that \(x = \frac{-10}{-7}\). We need to recognize how to handle negative signs in fractions. When dividing two negative numbers, the result is positive because two negatives make a positive.
Therefore, \(x = \frac{10}{7}\), which is a rational number. Any time we end up with fractions, it's important to check if they can be further simplified. In this case, \(\frac{10}{7}\) is already in its simplest form.

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Most popular questions from this chapter

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