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Chapter 6: Problem 11

Solve and check each equation. \(-8 x=2\)

Short Answer

Expert verified
The solution to the equation -8x=2 is \(x = -\frac{1}{4}\).

Step by step solution

01

Isolate x by dividing by -8

The objective of this step is to isolate x. To achieve this, you divide both sides of the equation \(-8 x = 2\) by \(-8\). The result is: \(x = - \frac{2}{8}\).
02

Simplify the fraction

Simplify the fraction -2/8 and you obtain \(x = -\frac{1}{4}\).
03

Check the solution

Substitute \(x = -\frac{1}{4}\) back into the original equation to verify it. This gives \(-8(-\frac{1}{4}) = 2\), which simplifies to \(2 = 2\), confirming that \(x = -\frac{1}{4}\) is indeed the solution.

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Key Concepts

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

Solving Equations
When faced with an algebraic equation, the goal is to determine the value of the unknown variable that makes the equation true. This process is known as "solving equations." In our example, we start with the equation \(-8x = 2\). To solve it:
  • We need to perform operations that simplify the equation, ultimately isolating the variable (here, \(x\)).
  • We strive to simplify both sides of the equation without altering its equality.
This involves using inverse operations to carefully undo any addition, subtraction, multiplication, or division present in the equation. It's like peeling away layers until we find the core value of the variable. Remember that whatever operation you perform on one side, you must also perform on the other to maintain balance, much like a balanced scale.
Isolating Variables
To find the value of the variable \(x\), we need to isolate it. Isolating a variable means getting the variable all by itself on one side of the equation. Here's how it was done:- **Divide by the Coefficient**: The equation is \(-8x = 2\). Here, \(-8\) is the coefficient of \(x\). To isolate \(x\), divide both sides by \(-8\) to neutralize the coefficient. Doing this, you end up with \(x = -\frac{2}{8}\).
  • Division by a negative number also reverses the sign of both sides.
  • This step ensures that \(x\) stands alone on one side, while a numerical value remains on the other.
By keeping \(x\) on one side and a number on the other, you simplify the problem to find what \(x\) equals.
Simplifying Fractions
Once we have isolated our variable and expressed it as a fraction, simplifying this fraction helps make our solution clearer. Simplifying fractions involves reducing them to their lowest possible terms.- **Simplification Process**: Start with the fraction \(-\frac{2}{8}\). Identify the greatest common divisor (GCD) of the numerator and the denominator. Since both 2 and 8 can be divided by 2, this is the GCD.- **Reduce the Fraction**: Divide both the numerator and the denominator by their GCD (2 in this case), resulting in \(-\frac{1}{4}\).
  • Simplifying makes numbers easier to understand and compare.
  • It ensures that the fraction is expressed in its simplest, most refined form.
After simplification, you are left with \(x = -\frac{1}{4}\), a cleaner and more understandable solution.

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Most popular questions from this chapter

Solve the equations in Exercises 53-72 using the quadratic formula. \(3 x^{2}=5 x-1\)

Factor the trinomials or state that the trinomial is prime. Check your factorization using FOIL multiplication. \(x^{2}-2 x-15\)

The formula $$ N=\frac{t^{2}-t}{2} $$ describes the number of football games, \(N\), that must be played in a league with t teams if each team is to play every other team once. Use this information to solve Exercises 83-84. If a league has 45 games scheduled, how many teams belong to the league, assuming that each team plays every other team once?

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The radicand of the quadratic formula, \(b^{2}-4 a c\), can be used to determine whether \(a x^{2}+b x+c=0\) has solutions that are rational, irrational, or not real numbers. Explain how this works. Is it possible to determine the kinds of answers that one will obtain to a quadratic equation without actually solving the equation? Explain.
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