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Chapter 2: Problem 108

Solve each equation. Be sure to check each answer. $$\frac{y}{16}=-5$$

Short Answer

Expert verified
The solution to the equation \(\frac{y}{16} = -5\) is \(y = -80\).

Step by step solution

01

Isolate y

We need to isolate \(y\) on one side of the equation. We can do this by multiplying both sides of the equation by 16. This is because when we multiply \(\frac{y}{16}\) by 16, it cancels out the division by 16, leaving just \(y\) on one side. This gives us: \(y = -5 \times 16\).
02

Solve for y

Next, we multiply -5 by 16 to obtain the value of \(y\). This is \(y = -80\)
03

Verify the solution

Finally, we can verify this solution by substituting \(y = -80\) back into the original equation. This gives us \(\frac{-80}{16}=-5\), which simplifies to -5 = -5. Since both sides of the equation are equal, this confirms that \(y = -80\) is indeed the correct solution.

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Key Concepts

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

Solving Equations
Solving equations is the process of finding the value of an unknown variable that makes the equation true. In algebra, this often involves finding what number a variable like \( y \) represents. Equations are like balanced scales, and the goal is to keep them balanced as you figure out what each variable stands for.
To solve equations, follow these steps:
  • Identify the variable that needs solving.
  • Perform operations that will isolate this variable on one side of the equation.
  • Ensure you do the same operation to both sides of the equation to keep it balanced.

In our example, the equation is \( \frac{y}{16} = -5 \). Solving it means manipulating the equation so \( y \) is by itself on one side.
Isolating Variables
Isolating a variable means arranging an equation so that the variable appears by itself on one side. This is crucial because it allows you to clearly see what value the variable must have.
To isolate \( y \) in the equation \( \frac{y}{16} = -5 \), you need to undo the division that involves \( y \).
Here's how it's done:
  • The equation starts as \( \frac{y}{16} = -5 \).
  • Multiply both sides by 16 to cancel out the division.
  • After multiplication, you're left with \( y = -5 \times 16 \).
  • This simplifies to \( y = -80 \).

Taking these steps systematically ensures you handle the isolation accurately and maintain the balance in your equation.
Verification of Solutions
Verification of a solution is just as important as solving the equation itself. Verifying helps confirm that the value you've found for the variable truly satisfies the original equation.
Here's how to verify:
  • Take the solution you found鈥攊n this case, \( y = -80 \).
  • Substitute it back into the original equation \( \frac{y}{16} = -5 \).
  • Substituting gives \( \frac{-80}{16} = -5 \).
  • Simplify the left side, and check if it equals the right side.
  • If both sides are equal, your solution is correct.

In our example, \( \frac{-80}{16} = -5 \) simplifies to \( -5 = -5 \), confirming that \( y = -80 \) is indeed a valid solution. Verification adds a layer of assurance that the solution is not just mathematically correct but logically sound.

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Most popular questions from this chapter

The volume \(V\) of a cone is given by the formula \(V=\frac{1}{3} \pi r^{2} h .\) Solve the formula for \(h\), and then calculate the height \(h\) if \(V\) is \(36 \pi\) cubic inches and the radius \(r\) is 6 inches.

Express each solution as an inequality. A student has test scores of \(68,75,\) and 79 points. What must she score on the fourth exam to have an average score of at least 80 points?

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Solve each application involving percents. Sales tax on a \(\$ 12\) compact disc is \(\$ 0.72 .\) At what rate is sales tax computed?

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