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Chapter 8: Problem 170

Solve the linear equation using the general strategy. $$3(10+5 r)=0$$

Short Answer

Expert verified
r = -2

Step by step solution

01

Distribute the Constant

Distribute the constant 3 to both terms inside the parentheses. This means multiplying 3 by 10 and 3 by 5r.
02

Rewrite the Equation

Rewrite the equation after distributing the 3: \[ 30 + 15r = 0 \]
03

Isolate the Variable Term

Subtract 30 from both sides to isolate the term with the variable r: \[ 15r = -30 \]
04

Solve for r

Divide both sides of the equation by 15 to solve for r: \[ r = \frac{-30}{15} \]
05

Simplify the Fraction

Simplify the fraction: \[ r = -2 \]

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Key Concepts

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

Distributive Property
The distributive property is a key technique in solving linear equations. It allows us to break down complex expressions into simpler parts. In our exercise, the original equation is \[ 3(10+5r) = 0 \].
To use the distributive property, we multiply each term inside the parentheses by the number outside. This turns \[ 3(10+5r) \] into \[ 3 \times 10 + 3 \times 5r \].
After performing these multiplications, we get a simplified equation: \[ 30 + 15r = 0 \].
With the distributive property, equations can be made simpler and easier to solve step by step.
Isolating the Variable
Isolating the variable is an important step to solving equations, especially when dealing with linear equations like the one in this example. After using the distributive property in our exercise, we reach \[ 30 + 15r = 0 \].
Our goal now is to isolate the term with the variable r. To do this, we need to get rid of the constant term on the same side as r. We subtract 30 from both sides: \[ 30 + 15r - 30 = 0 - 30 \], which simplifies to \[ 15r = -30 \].
Now, the term with the variable r is isolated, making it much easier to solve for the value of r.
Simplifying Fractions
Simplifying fractions is another essential skill in solving linear equations. After isolating the variable term in the exercise, we have \[ 15r = -30 \].
To solve for r, we divide both sides by 15: \[ r = \frac{-30}{15} \].
The next step is simplifying the fraction. We divide the numerator (-30) and the denominator (15) by their greatest common divisor, which is 15. This gives us: \[ r = \frac{-30 \div 15}{15 \div 15} = \frac{-2}{1} = -2 \].
Simplifying fractions helps make the solution clear and ensures the answer is in its simplest form.

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Most popular questions from this chapter

In the following exercises, solve each equation. $$ 4 y-8=16 $$

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In the following exercises, solve each equation. $$ -5(2 x+1)=45 $$

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