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Chapter 3: Problem 33

Solve each equation. $$14 y+15=-33$$

Short Answer

Expert verified
The solution to the equation is \(y = -\frac{24}{7}\).

Step by step solution

01

Subtract the Constant from Both Sides

We start with the equation:\[14y + 15 = -33\]To isolate the term with the variable \(y\), subtract 15 from both sides:\[14y + 15 - 15 = -33 - 15\]This simplifies to:\[14y = -48\]
02

Divide by the Coefficient of y

Now, we need to solve for \(y\) by dividing both sides of the equation by 14, the coefficient of \(y\):\[\frac{14y}{14} = \frac{-48}{14}\]Simplifying the fractions gives us:\[y = -\frac{24}{7}\]
03

Simplify the Answer if Needed

The fraction \(-\frac{48}{14}\) simplifies into the lowest terms:\[y = -\frac{24}{7}\]Thus, the simplified answer is:\[y = -\frac{24}{7}\]

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Key Concepts

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

Solving Equations
The process of solving equations involves finding the value of the variable that makes the equation true. In this problem, the variable is \( y \), and the equation we want to solve is \( 14y + 15 = -33 \). To solve equations effectively:
  • Start by identifying the terms on both sides of the equation.
  • Try to isolate the variable on one side by using inverse operations.
  • Check your work by substituting the solution back into the original equation to verify your results.
In our case, we isolate \( y \) by first eliminating the constant on the same side as \( y \), followed by dividing both sides by the coefficient of \( y \). This systematic approach helps in finding the value of \( y \) that satisfies the equation.
Algebraic Manipulation
Algebraic manipulation allows us to rearrange and solve equations by using mathematical operations like addition, subtraction, multiplication, and division. In this exercise:
  • We subtracted 15 from both sides of the equation to shift the constant term to the other side. This gives us the equation \( 14y = -48 \).
  • The key to algebraic manipulation is maintaining balance, meaning whatever operation you do to one side should also be done to the other side.
  • Always perform one operation at a time to avoid errors and simplify each step to keep the equation as clear as possible.
By carefully applying these operations, we transform the original statement into a simpler one, making it easier to solve for \( y \).
Simplifying Fractions
Simplifying fractions is an essential skill in algebra. When you have an equation where fractions appear, like our solution \( y = -\frac{48}{14} \), simplifying is crucial. Here's how:
  • Factor both the numerator and the denominator to find their greatest common divisor (GCD).
  • Divide both by their GCD. In this instance, \( 48 \) and \( 14 \) have a GCD of 2. After dividing, the fraction simplifies to \( -\frac{24}{7} \).
  • Always check if the fraction can be reduced further to its simplest form.
Simplifying fractions makes equations easier to interpret and solve accurately. It conveys the most compact and readable form of the solution, which is often necessary in mathematics.

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Most popular questions from this chapter

Solve each equation. $$\frac{x+7}{3}-\frac{x+9}{6}=\frac{5}{9}$$

Consider these two solutions: $$ \begin{array}{rrr} 3(x+2)=9 & 3(x-4)=7 \\ \frac{3(x+2)}{3}=\frac{9}{3} & 3 x-12=7 \\ x+2=3 & 3 x=19 \\ x=1 & x=\frac{19}{3} \end{array} $$ Are both of these solutions correct? Comment on the effectiveness of the two different approaches.

Find three consecutive whole numbers sach that twice the sum of the two smallest numbers is 10 more than three times the largest number.

The sum of two numbers is 53 . If three times the smaller number is 1 less than the larger number, find the numbers.

Mrs. Nelson had to wait 4 minutes in line at her bank's automated teller machine. This was 3 minutes less than one-half of the time she waited in line at the grocery store. How long in minutes did she wait in line at the grocery store?
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