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Chapter 4: Problem 63

Solve the equations and inequalities. $$3(y+2)+\frac{y+3}{5}=\frac{9 y+8}{2}$$

Short Answer

Expert verified
The solution is \( y = 2 \).

Step by step solution

01

- Clear the Fraction

Multiply every term by the least common multiple (LCM) of the denominators, which is 10, to eliminate the fractions: \[10 \times 3(y+2) + 10 \times \frac{y+3}{5} = 10 \times \frac{9y+8}{2}\]
02

- Simplify the Equation

Distribute the 10 to clear the fractions: \[30(y+2) + 2(y+3) = 5(9y+8)\]This simplifies to: \[30y + 60 + 2y + 6 = 45y + 40\]
03

- Combine Like Terms

Combine like terms on both sides of the equation: \[32y + 66 = 45y + 40\]
04

- Isolate the Variable

Subtract 32y from both sides to isolate the variable term: \[66 = 13y + 40\]Then, subtract 40 from both sides: \[26 = 13y\]
05

- Solve for y

Divide both sides by 13 to get the value of y: \[y = 2\]

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Key Concepts

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

Linear Equations
A linear equation is an equation in which the highest power of the variable is one. In other words, it graphs to a straight line. Linear equations can often be identified by their standard form: \[ax + b = c\] where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. They frequently appear in the form of:
  • Simple one-variable equations, like \(3x + 5 = 2\)
  • More complex forms involving fractions or multiple terms, such as \(2x + \frac{3}{4} = 5 - x\)
To solve linear equations, one typically combines like terms, uses inverse operations, and aims to isolate the variable. This allows for finding the precise value that satisfies the equation.
Fractions
Fractions represent parts of a whole number and are written as \(\frac{a}{b}\), where \(a\) is the numerator and \(b\) is the denominator. Fractions can make solving equations a bit trickier, as they introduce additional steps, such as finding a common denominator.
When working with fractions in equations:
  • Look for opportunities to clear the fraction by multiplying every term by the least common multiple (LCM) of the denominators.
  • Simplify fractions by reducing them to their lowest form.
  • Combine fractional terms by ensuring they have a common denominator.
For example, in the equation \(3(y+2) + \frac{y+3}{5} = \frac{9y+8}{2}\), the LCM of 2 and 5 is 10. Multiplying every term by 10 clears the fractions, simplifying the equation and making it easier to solve.
Variable Isolation
Variable isolation involves manipulating an equation to get the variable alone on one side. This is key to solving for the variable's value. For example, consider the steps to solve for \(y\) in the equation:
\[30(y+2) + 2(y+3) = 5(9y+8)\] First, distribute occurrences and combine like terms: \[30y + 60 + 2y + 6 = 45y + 40\] Which simplifies to: \[32y + 66 = 45y + 40\] Next, subtract \(32y\) from both sides: \[66 = 13y + 40\] Then, isolate \(y\) by subtracting 40 from both sides and dividing by 13: \[26 = 13y\] \[y = 2\]
Simplification
Simplification of equations means breaking them down step-by-step until they are easy to solve. This involves:
  • Applying basic arithmetic operations, like addition and subtraction, to combine like terms.
  • Using distributive properties to clear parentheses and simplify expressions.
  • Reducing fractions by multiplying through with the least common multiple (LCM).
In our example, multiple steps of simplification were applied:
First, clearing the fraction: \[30(y+2) + 2(y+3) = 5(9y+8)\] Then, combining and simplifying: \[30y + 60 + 2y + 6 = 45y + 40\] Finally, further reducing and solving: \[32y + 66 = 45y + 40\]
Effective simplification makes it clearer and more manageable to isolate the variable and solve the equation.

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Most popular questions from this chapter

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. How much pure antifreeze should be added to a radiator that contains 10 gallons of a \(30 \%\) antifreeze solution to produce a \(50 \%\) antifreeze solution? [Hint: Pure antifreeze is \(100 \% \text { antifreeze. }]\)

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. The length of a rectangle is 1 less than twice the width. If the width is 1 more than one seventh of the perimeter, find the dimensions of the rectangle.

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. How many pounds of coffee beans selling at \(\$ 6.65 / 1 b\) should be mixed with how many pounds of coffee beans selling for \(\$ 4.25 / 1 b\) to produce a 24 -lb mixture that should sell for \(\$ 5.75 / 1 b ?\)

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. Lam takes five exams and has a semester average of \(92 .\) Four of his exam scores were \(79,89,94,\) and \(98 .\) Find his fifth exam score.

Set up an equation or inequality and solve the problem. Be sure to indicate clearly what quantity your variable represents. Round to the nearest tenth where necessary. Lisa has scores of \(79,83,88,\) and 91 on four class exams. What score must she get on her fifth exam so that her exam average is at least \(85 ?\)
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