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Chapter 5: Problem 10

Solve the equations. $$ 7-3 y=-17 $$

Short Answer

Expert verified
Answer: The value of \(y\) in the given linear equation is \(8\).

Step by step solution

01

Isolate the term with the variable y

In the given equation, \(7 - 3y = -17\), we'll first isolate the term with the variable \(y\) by moving the constant term \(7\) to the other side of the equation. To do this, subtract \(7\) from both sides: $$ 7 - 3y - 7 = -17 - 7 $$ This simplifies to: $$ -3y = -24 $$
02

Solve for the variable y

Now that the term with the variable \(y\) is isolated, to solve for \(y\), divide both sides of the equation by the coefficient of \(y\), which is \(-3\): $$ \frac{-3y}{-3} = \frac{-24}{-3} $$ This simplifies to: $$ y = 8 $$ The solution to the given linear equation is \(y = 8\).

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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 a foundational skill in algebra. At its core, solving an equation means finding the value of the variable that makes the equation true. For example, in the equation \(7 - 3y = -17\), we're tasked with finding the value of \(y\) that balances both sides.
The process involves several steps that strategically simplify the equation, making it easier to determine the variable's value. While some equations are straightforward, others require a keen eye for details.
Below are important general principles involved in solving equations:
  • Identify terms that can be combined or simplified.
  • Use inverse operations, such as addition and subtraction or multiplication and division, to cancel out terms.
  • Maintain the balance of the equation by performing the same operation on both sides.
By understanding these steps, you can solve not only simple but also more complex equations.
Isolation of Variables
Isolation of variables is a crucial technique in solving equations. It refers to the process of rearranging the equation so that the variable is on one side by itself. For example, in the equation \(7 - 3y = -17\), the first step was to isolate the term \(-3y\).
To achieve this, follow these general steps:
  • Identify the variable and the terms associated with it (e.g., \(-3y\) in this example).
  • Move other terms to the opposite side by performing inverse operations. Here, subtract \(7\) from both sides to eliminate the constant on the left-hand side.
  • Reassess the equation to ensure only the term containing the variable remains on one side.
This isolation makes it clear what further steps are needed to solve for the variable.
Algebraic Manipulation
Algebraic manipulation is the art of rearranging equations using mathematical operations. It's vital for solving equations efficiently. In the example \(-3y = -24\), we applied algebraic manipulation to simplify and solve for \(y\).
Here's a closer look at what this involves:
  • Use cancellation by dividing both sides by the coefficient of the variable. If the equation is \(-3y = -24\), divide by -3 to isolate \(y\).
  • Simplify the expression post-manipulation. Ensure all calculations are executed correctly to avoid errors.
  • Aim for the simplest possible form of the equation where the variable is completely isolated.
Efficient algebraic manipulation can simplify complex problems, making them much more approachable.

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Most popular questions from this chapter

\(f(t)=2 t+7\). Does the equation have no solution, one solution, or an infinite number of solutions? $$ f(t)=-f(t) $$

Give the slope and \(y\) -intercept for the graphs of the functions. $$ f(x)=\pi x $$

\(f(t)=2 t+7\). Does the equation have no solution, one solution, or an infinite number of solutions? $$ f(t)+f(3 t)-2 f(2 t)=0 $$

Decide for what value(s) of the constant \(A\) (if any) the equation has (a) The solution \(x=0\) (b) A positive solution in \(x\) (c) No solution in \(x\). $$ 3 x+5=A $$

\(f(t)=2 t+7\). Does the equation have no solution, one solution, or an infinite number of solutions? $$ f(t)=f(t+1)-2 $$
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