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Chapter 2: Problem 6

Solve the equation. $$6(2 y+3)-3(y-5)=0$$

Short Answer

Expert verified
The solution is \(y = -\frac{11}{3}\).

Step by step solution

01

Distribute the terms inside the parentheses

Distribute the 6 into the first set of parentheses and -3 into the second set of parentheses. Distributing 6: \[6(2y + 3) = 12y + 18\] Distributing -3: \[-3(y - 5) = -3y + 15\] Now, the equation becomes: \[12y + 18 - 3y + 15 = 0\]
02

Combine like terms

Combine the like terms on the left side of the equation. Combine the \(y\) terms: \[12y - 3y = 9y\] Combine the constant terms: \[18 + 15 = 33\] Now our equation is: \[9y + 33 = 0\]
03

Isolate the variable term

Subtract 33 from both sides of the equation to isolate the variable term. \[9y + 33 - 33 = 0 - 33\] This simplifies to: \[9y = -33\]
04

Solve for y

Divide both sides by 9 to solve for \(y\). \[\frac{9y}{9} = \frac{-33}{9}\] Simplify the right side: \[y = -\frac{33}{9}\] Which reduces further to: \[y = -\frac{11}{3}\]

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Key Concepts

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

Distributive Property
In the given problem, applying the distributive property is essential to simplify expressions by removing the parentheses. The distributive property states that a term outside the parentheses multiplies each term inside the parentheses. This is expressed mathematically as \[ a(b+c) = ab + ac. \]
Following this property, in our equation \[6(2y + 3) - 3(y - 5), \] you distribute 6 and -3 just like instructed. Thus, we convert the expression to \[12y + 18 - 3y + 15 = 0. \]
This demonstrates the power of the distributive property in expanding and simplifying expressions, essential for solving equations.
Combining Like Terms
Once the distributive property has been applied, the next step is to combine like terms. Like terms are terms in an algebraic expression that have the same variable raised to the same power. Simplifying these makes the equation more manageable.
In the expression \[12y + 18 - 3y + 15, \] the like terms are the terms containing the same variable, namely \[12y \] and \[-3y, \] which can be combined as \[9y, \] and the constant terms \[18 \] and \[15, \] which sum up to \[33.\]
After combining, the equation simplifies to \[9y + 33 = 0, \] which is simpler and closer to finding the solution.
Isolating Variables
The goal of isolating the variable is to have the variable term alone on one side of the equation. This allows us to determine the value of the variable easily. In our simplified equation \[9y + 33 = 0, \] you can isolate the variable term \[9y \] by subtracting \[33 \] from both sides, leading to \[9y = -33.\]
This action effectively leaves \[9y \] by itself on one side and prepares us for the final step to solve for \[y.\] Isolation is crucial because it transforms the equation into a form that is ready for final solutions.
Fractions in Equations
Finally, when solving for the variable, you may encounter fractions, as we see in this problem when we divide both sides by 9: \[ \frac{9y}{9} = \frac{-33}{9}. \] This step reduces further to \[ y = -\frac{11}{3}.\]
Understanding fractions in equations is vital, as they can often appear when dividing coefficients to isolate the variable. Fraction reduction involves finding the greatest common divisor (GCD) of the numerator and denominator, simplifying them to their lowest terms, which in this case makes \[ -\frac{33}{9}\] become \[-\frac{11}{3}. \] Being comfortable with fractions is fundamental when solving equations, as they represent exact division results.

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Most popular questions from this chapter

Archeologists can determine the height of a human without having a complete skeleton. If an archeologist finds only a humerus, then the height of the individual can be determined by using a simple linear relationship. (The humerus is the bone between the shoulder and the elbow.) For a female, if \(x\) is the length of the humerus (in centimeters), then her height \(h\) (in centimeters) can be determined using the formula \(h=65+3.14 x\). For a male, \(h=73.6+3.0 x\) should be used. (a) A female skeleton having a 30 -centimeter humerus is found. Find the woman's height at death. (b) A person's height will typically decrease by \(0.06\) centimeter each year after age 30 . A complete male skeleton is found. The humerus is 34 centimeters, and the man's height was 174 centimeters. Determine his approximate age at death.

The height \(h\) (in feet) of the cloud base can be estimated using \(h=227(T-D)\), where \(T\) is the ground temperature and \(D\) is the dew point. (a) If the temperature is \(70^{\circ} \mathrm{F}\) and the dew point is \(55^{\circ} \mathrm{F}\), find the height of the cloud base. (b) If the dew point is \(65^{\circ} \mathrm{F}\) and the cloud base is 3500 feet, estimate the ground temperature.

In a certain medical test designed to measure carbohydrate tolerance, an adult drinks 7 ounces of a \(30 \%\) glucose solution. When the test is administered to a child, the glucose concentration must be decreased to \(20 \%\). How much \(30 \%\) glucose solution and how much water should be used to prepare 7 ounces of \(20 \%\) glucose solution?

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A large grain silo is to be constructed in the shape of a circular cylinder with a hemisphere attached to the top (see the figure). The diameter of the silo is to be 30 feet, but the height is yet to be determined. Find the height \(h\) of the silo that will result in a capacity of \(11,250 \pi \mathrm{ft}^{3}\).
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