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Chapter 2: Problem 68

Solve each equation, and check your solution. $$ 2(2-3 r)=-5(r-3) $$

Short Answer

Expert verified
r = -11

Step by step solution

01

- Distribute the Constants

Distribute the constants in both sides of the equation to eliminate the parentheses: 2(2-3r) = -5(r-3) becomes 4 - 6r = -5r + 15.
02

- Rearrange to Isolate the Variable

Rearrange the equation to gather all terms containing the variable r on one side: 4 - 6r + 5r = 15 results in 4 - r = 15.
03

- Solve for the Variable

Solve for r by isolating it on one side of the equation: 4 - 15 = r produces -11 = r, or r = -11.
04

- Check the Solution

Substitute -11 for r in the original equation to verify the solution: 2(2 - 3(-11)) = -5(-11 - 3), which simplifies to 2(2 + 33) and -5(-14), resulting in 2(35) = -5(-14) or 70 = 70, confirming the solution.

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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 essential in solving linear equations with parentheses. It allows us to eliminate the parentheses and simplify the equation.
This property states that for any numbers a, b, and c, the equation a(b + c) equals ab + ac.
In our original exercise, we distributed the constants as follows:
\[2(2-3r) = -5(r-3)\]
Applying the distributive property, we get:
\[2 * 2 + 2 * (-3r) = -5 * r + -5 * (-3)\]
This simplifies to:
\[4 - 6r = -5r + 15\]
By distributing the constants, we transformed a more complex equation into a simpler one, making it easier to solve.
Isolating the Variable
To solve for the variable r, we need to isolate it on one side of the equation. This involves re-arranging terms and performing arithmetic operations.
From our equation after applying the distributive property:
\[4 - 6r = -5r + 15\]
We start by bringing all terms involving r to one side and constants to the other. To achieve this, we add 5r to both sides:
\[4 - 6r + 5r = 15\]
This simplifies to:
\[4 - r = 15\]
Next, we want to isolate r by subtracting 4 from both sides:
\[4 - 4 - r = 15 - 4\]
This results in:
\[-r = 11\]
Finally, to solve for r, we multiply both sides by -1:
\[r = -11\].
Isolating the variable helps us get to the solution step-by-step.
Checking Solutions
After solving for the variable, it's crucial to check the solution. This confirms the accuracy of our calculations.
In our exercise, we found \(r = -11\). We substitute this value back into the original equation to verify it:
\[2(2-3(-11)) = -5(-11-3)\]
This simplifies to:
\[2(2 + 33) = -5(-14)\]
Further simplifying, we get:
\[2 * 35 = -5 * -14\] or \[70 = 70\]
Both sides of the original equation are equal, confirming our solution is correct.
This step ensures our solution is accurate and the equation is correctly solved.

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