/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Solve \(3 y-1=11\) for \(y\).... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 1

Solve \(3 y-1=11\) for \(y\).

Short Answer

Expert verified
Answer: The value of \(y\) in the linear equation \(3y-1=11\) is 4.

Step by step solution

01

Add 1 to both sides of the equation

First, we will add 1 to both sides of the equation in order to eliminate the constant term on the left side of the equation. This gives us: \(3y - 1 + 1 = 11 + 1\) Which simplifies to: \(3y = 12\)
02

Divide both sides of the equation by 3

Now that we have isolated the term with the variable \(y\), we will divide both sides of the equation by the coefficient of \(y\), which is 3. This allows us to solve for \(y\): \(\frac{3y}{3} = \frac{12}{3}\) This simplifies to: \(y = 4\) Therefore, the solution to the equation \(3y-1=11\) is \(y = 4\).

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Key Concepts

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

Solving Equations
When solving equations, your main goal is to find the value of the unknown variable that makes the equation true. In the problem, we want to find what number could replace \( y \) in the equation \( 3y-1=11 \) to make it accurate.
To achieve this, you'll manipulate the equation using algebraic rules to simplify it and isolate the variable. Always remember, the same operations must be applied to both sides of the equation to maintain balance. It's like keeping both sides of a scale even.
  • Identify the unknown variable you need to solve for.
  • Use logical steps for manipulating the equation.
  • Perform inverse operations to move terms as needed.
Solving equations is like unraveling a puzzle until you find what value fits perfectly for the variable.
Algebraic Manipulation
Algebraic manipulation involves performing operations with the numbers and variables in an equation. It's used to transform the equation into a more straightforward form where the solution is easier to see.
In our exercise, the initial equation is \( 3y-1=11 \). To solve this, we begin by performing algebraic manipulation. We notice a \(-1\) is subtracted from \( 3y \), and to undo this, we add \( 1 \) to both sides:

\[ 3y - 1 + 1 = 11 + 1 \]
This changes our equation to:
\[ 3y = 12 \]
This process involves:
  • Adding or subtracting terms to each side to eliminate constants adjacent to the variable.
  • Ensuring each step simplifies the equation further toward solution.
Remember, the key to algebraic manipulation is understanding how numbers and variables interact through operations.
Equation Isolation
Equation isolation is a critical skill when you want to solve for a specific variable. It means getting the variable alone on one side of the equation. Once isolated, you can plainly state the solution.
In our example, after simplifying the equation to \( 3y = 12 \), we aim to isolate \( y \) by eliminating its coefficient. This involves dividing each side of the equation by \( 3 \):

\[ \frac{3y}{3} = \frac{12}{3} \]
Which results in:
\[ y = 4 \]
Through www.equation isolation:
  • Identify the coefficient linked to the variable.
  • Apply division across the equation to remove the coefficient.
  • Confirm that the variable stands alone on one side of the equation.
The isolation process lets you directly interpret the solution, finding the variable's value that satisfied the original equation.

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Most popular questions from this chapter

For the following problems, solve the linear equations in two variables. $$ -x-y=0, \text { if } y=\frac{14}{3} $$

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Twenty-five percent of a number is \(12.32 .\) What is the number?

Translate the phrases or sentences to mathematical expressions or equations. Twice a quantity plus nine is equal to the quantity plus sixty.

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