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Chapter 1: Problem 5

\(\frac{3}{5}(y-5)=9\)

Short Answer

Expert verified
The solution to the equation is \(y=50\).

Step by step solution

01

Removing Fraction Multiplier

Start by isolating \(y\) in the equation. First, get rid of the fraction in front of the brackets. This can be done by multiplying both sides of equation by the reciprocal of \(\frac{3}{5}\), which is \(\frac{5}{3}\). This gives: \(\frac{5}{3} \cdot \frac{3}{5}(y-5) = \frac{5}{3} \cdot 9\). Thus, \(y-5 = 5 \cdot 9.\)
02

Solving for y

Next, simplify the equation by performing the multiplication on the right side, resulting in \(y-5=45\). After this, add 5 to both sides to isolate \(y\). So, \(y = 45 + 5\).
03

Final Result

Finally, add the numbers on the right side of the equation to get the value of \(y\). This gives: \(y = 50\).

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Key Concepts

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

Fraction Manipulation
When dealing with equations that have fractions, it's often easiest to eliminate the fraction first. This process usually involves multiplying both sides of the equation by the reciprocal of the fraction.

For example, in the equation \(\frac{3}{5}(y-5) = 9\), we have \(\frac{3}{5}\) as a fractional multiplier. To eliminate it, multiply both sides by its reciprocal, \(\frac{5}{3}\). This transforms the equation into:
  • \(\frac{5}{3} \cdot \frac{3}{5}(y-5) = \frac{5}{3} \cdot 9\)
  • which simplifies to \(y-5 = 15\)
Fraction manipulation helps simplify the equation, making it easier to solve. Always double-check to ensure the fraction is completely removed before proceeding to the next step.
Isolation of Variable
The goal of solving an equation is to isolate the variable on one side. This means you manipulate the equation until the variable you're solving for stands alone. In our exercise's case, it's about getting \(y\) by itself.

After eliminating the fraction, the equation might look something like \(y-5 = 15\). To isolate \(y\), you need to get rid of the \(-5\).

You can do this by adding 5 to both sides of the equation:
  • \(y - 5 + 5 = 15 + 5\)
  • Simplifying gives \(y = 20\)
In isolation, ensure every operation you do is equally applied on both sides of the equation. This maintains the balance, leading you directly to the answer.
Algebraic Operations
Algebraic operations are the actions you perform to simplify or solve an equation. These include addition, subtraction, multiplication, and division applied to both numbers and variables.

In our example, we performed several steps of algebraic operations.
  • First, we multiplied by the reciprocal to eliminate a fraction.
  • Next, we simplified the multiplication to clear the brackets.
  • Finally, we used addition to isolate the variable.
Always handle algebraic operations carefully to maintain the equation's balance. Double-check each step, ensuring that your operations align with the equation's rules. This accuracy ensures successful problem-solving.

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Most popular questions from this chapter

Solve the inequality. Express your answer in interval notation. $$\frac{1}{3}(x+1)

This set of exercises will draw on the ideas presented in this section and your general math background. Find the intersection of the lines \(x+y=2\) and \(x-y=1\) You will have to first solve for \(y\) in both equations and then use the methods presented in this section. (This is an example of a system of linear equations, a topic that will be explored in greater detail in a later chapter.)

Pricing Tickets Sherman is planning to bring in a jazz group of four musicians for a fund-raising concert at Grand State University. The jazz group charges \(\$ 500\) for an appearance, and dinner will be provided to the musicians at a cost of \(\$ 20\) each. In addition, the musicians will be reimbursed for mileage at a rate of \(\$ 0.30\) permile. The group will be traveling a total of 160 miles. A ticket for the concert will be priced at \(\$ 8 .\) How many people must attend the concert for the university to break even?

The piecewise-defined function given below is known as the characteristic function, \(C(x) .\) It plays an important role in advanced mathematics. $$C(x)=\left\\{\begin{array}{ll}0, & \text { if } x \leq 0 \\\1, & \text { if } 0

Graph the function by hand. $$f(x)=\left\\{\begin{array}{ll} 3, & x \leq-1 \\ -x+2, & x>-1 \end{array}\right.$$
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