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Chapter 2: Problem 41

Solve each formula for the indicated letter. $$ 3 x-5 y=8, \text { for } y $$

Short Answer

Expert verified
y = \frac{3x - 8}{5}

Step by step solution

01

Add 5y to both sides

Start by rearranging the equation. Add 5y to both sides to get rid of the negative sign. The equation becomes: 3x = 8 + 5y.
02

Subtract 8 from both sides

Next, subtract 8 from both sides of the equation. This gives: 3x - 8 = 5y.
03

Divide both sides by 5

Finally, isolate y by dividing both sides by 5. The equation becomes: \[ y = \frac{3x - 8}{5} \].

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Key Concepts

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

linear equations
When we talk about linear equations, we refer to equations where the highest degree of any variable is one. These are the simplest kind of equations and form the foundation for more complex algebraic concepts. The general form of a linear equation is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. For the provided problem, our equation is \(3x - 5y = 8\). Linear equations often represent lines when graphed on a coordinate plane, making them very useful in both algebra and geometry. Understanding linear equations is essential because it paves the way for learning how to solve more complex mathematical problems.
isolation of variables
Isolation of variables means solving for one variable in terms of the other variables present in the equation. This is an essential problem-solving technique in algebra. Here's a breakdown of how to isolate a variable: In our exercise, we need to solve for \(y\).
  • Step 1: Add or subtract terms on both sides to begin isolating \(y\). Here, we add \(5y\) to both sides. The equation transforms into \(3x = 8 + 5y\).
  • Step 2: Continue simplifying so that \(y\) is on one side of the equation. We subtract 8 from both sides to get: \(3x - 8 = 5y\).
  • Step 3: Divide by the coefficient of \(y\) to get the final isolated \(y\) term: \(y = \frac{3x - 8}{5}\).
By following these steps, we effectively isolate the variable we need.
rearranging equations
Rearranging equations is a key skill that simplifies finding solutions in algebra. It involves manipulating the given equation to express one variable in terms of others. With practice, this becomes second nature. To rearrange an equation, keep these tips in mind:
  • Use inverse operations to move terms from one side to the other. This often means adding, subtracting, multiplying, or dividing.
  • Maintaining the equality is crucial. Everything you do to one side must be done to the other.
  • Try to make one side as simple as possible by combining like terms or factoring.
In the exercise, we saw this:
  • First, transformed \(3x - 5y = 8\) into \(3x = 8 + 5y\).
  • Then, simplified it to \( 3x - 8 = 5y\).
  • Finally, isolated \(y\) to get \( y = \frac{3x - 8}{5}\).
Practicing rearranging equations ensures you can tackle diverse algebra problems with ease.

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Most popular questions from this chapter

Use an inequality and the five-step process to solve each problem. The average price of a movie ticket can be estimated by the equation \(P=0.169 Y-333.04,\) where \(Y\) is the year and \(P\) is the average price, in dollars. For what years will the average price of a movie ticket be at least \(\$ 7 ?\) (Include the year in which the \(\$ 7\) ticket first occurs.

Are the inequalities \(x>-3\) and \(x \geq-2\) equivalent? Why or why not?

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Match the sentence with one of the following: $$ a

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