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Chapter 2: Problem 13

Solve each equation. \(-6 y-4=16\)

Short Answer

Expert verified
\(y = -\frac{10}{3}\).

Step by step solution

01

Identify the Equation

The equation we need to solve is \(-6y - 4 = 16\).Our goal is to isolate the variable \(y\).
02

Add 4 to Both Sides

To start isolating \(y\), we first remove the constant term on the left side by adding 4 to both sides of the equation:\[-6y - 4 + 4 = 16 + 4.\]This simplifies to:\[-6y = 20.\]
03

Divide Both Sides by -6

Now, to isolate \(y\), we need to divide both sides of the equation by -6:\[y = \frac{20}{-6}.\]Simplifying \(\frac{20}{-6}\) gives us:\[y = -\frac{10}{3}.\]

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Key Concepts

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

Algebraic Manipulation
When solving linear equations, algebraic manipulation is a crucial skill. It involves rearranging and simplifying terms to make equations easier to solve. In the original problem, \(-6y - 4 = 16\), we begin with algebraic manipulation by adding 4 to both sides, which balances the equation. This technique of adding or subtracting the same value from both sides is fundamental. Here鈥檚 why it's important:
  • It ensures the equation remains balanced, similar to keeping scales even.
  • It helps in gradually moving towards isolating the variable, which is our ultimate goal.
Remember, each step of manipulation should maintain the equality, which is key to reaching a solution accurately.
Isolation of Variable
Isolating the variable is at the heart of solving equations. Our aim is to get the variable alone on one side of the equation.In \(-6y = 20\), the next step is to isolate \(y\). This means getting \(y\) alone by itself.Here鈥檚 how you can effectively isolate a variable:
  • Identify any coefficients (numbers multiplied by the variable) and constant terms.
  • Use inverse operations to eliminate the numbers. For instance, if the variable is multiplied by \(-6\), divide by \(-6\) to eliminate it.
In our example, dividing both sides by \(-6\) isolates \(y\), demonstrating that calculating such operations requires careful attention to signs (negative or positive) to maintain the equation's integrity.
Simplifying Fractions
After isolating the variable, you oftentimes must simplify the result. Simplifying fractions makes the solution more elegant and easier to understand.In the equation, \(y = \frac{20}{-6}\), simplify this fraction by finding common factors. Notice that both 20 and 6 are divisible by 2:\[\frac{20}{-6} = \frac{10}{-3}\]Here's a quick guide to simplify fractions:
  • Find the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that divides both without a remainder.
  • Divide both the numerator and the denominator by their GCD.
After simplifying, \(y = -\frac{10}{3}\) becomes your cleaner, final answer. Proper simplification ensures clarity and accuracy in any mathematical solution.

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Most popular questions from this chapter

The temperatures for a 24-hour period ranged between \(-4^{\circ} \mathrm{F}\) and \(23^{\circ} \mathrm{F}\), inclusive. What was the range in Celsius degrees? \(\left(\right.\) Use \(F=\frac{9}{5} C+32\). \()\)

Solve each equation. $$ |x+1|=|x-1| $$

Solve each equation and inequality by inspection. $$ |x+4|<0 $$

How do you know by inspection that the solution set of the inequality \(x+3>x+2\) is the entire set of real numbers?

For Problems \(31-54\), solve each inequality. $$ |x+3|<5 $$
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