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Chapter 2: Problem 68

Solve each equation, and check the solution. $$ 9 r+2 r-7 r=68 $$

Short Answer

Expert verified
The solution is r = 17, and the solution is verified as correct.

Step by step solution

01

Combine Like Terms

Combine the like terms on the left side of the equation. Here, combine the terms involving the variable 'r': $$ 9r + 2r - 7r $$ $$ (9 + 2 - 7) r $$. This simplifies to:$$ 4r. $$ So, the equation now is: $$ 4r = 68 $$.
02

Solve for r

To solve for 'r', you need to isolate 'r' on one side of the equation. Divide both sides of the equation by 4: $$ r = \frac{68}{4} $$. So, $$ r = 17 $$.
03

Check the Solution

Substitute 'r' = 17 into the original equation to verify the solution: $$ 9(17) + 2(17) - 7(17) = 68 $$ Calculate each term: $$ 153 + 34 - 119 = 68 $$ Verify by simplification: $$ 187 - 119 = 68 $$ Finally: $$ 68 = 68 $$ The left side equals the right side, so the solution is correct.

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Key Concepts

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

Combining Like Terms
When solving linear equations, it's essential to simplify the equation as much as possible. This often starts with combining like terms. Like terms are terms that contain the same variable raised to the same power. For instance, in the equation given, \(9r + 2r - 7r\), all terms have the variable 'r'. This means they can be combined:
  • First, identify the like terms: \(9r\), \(2r\), and \(7r\).
  • Next, add or subtract the coefficients (the numbers in front of 'r'): \((9 + 2 - 7)r\).
  • This simplifies to \(4r\).
So, the simplified equation is \(4r = 68\). Combining like terms makes the equation easier to solve, as it reduces the number of terms you need to work with.
Isolating Variables
Once the equation is simplified, the goal is to isolate the variable. This means getting the variable by itself on one side of the equation. In our example, after combining like terms, we have \(4r = 68\). To isolate 'r':
  • Divide both sides of the equation by the coefficient of 'r', which is 4.
  • This gives \(r = \frac{68}{4}\).
  • Perform the division to find \(r = 17\).
Now 'r' is isolated and solved. This step is crucial in solving linear equations, as it allows us to find the value of the variable. Always ensure the variable is completely by itself on one side of the equation to verify the solution correctly.
Checking Solutions
After finding the solution to the equation, it's important to verify that the solution is correct. This involves substituting the found value back into the original equation and checking both sides. For the solution \(r = 17\):
  • Substitute \(r = 17\) back into the original equation: \(9(17) + 2(17) - 7(17) = 68\).
  • Calculate each term: \(153 + 34 - 119\).
  • Combine the results: \(153 + 34 - 119 = 187 - 119 = 68\).
Both sides of the equation are equal, confirming that the solution is correct. Checking solutions ensures that no mistake was made during the solving process and confirms the accuracy of your answer. It's a good habit to always check your work, especially when dealing with algebraic equations.

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