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Chapter 5: Problem 90

Solve and check. $$5 x+2 x=14$$

Short Answer

Expert verified
The solution to the equation \(5x + 2x = 14\) is \(x = 2\).

Step by step solution

01

Combine Like Terms

Given equation is \(5x + 2x = 14\). Combining like terms (i.e., the 'x' terms), you get \(7x = 14\).
02

Solve for 'x'

Now, to solve for 'x', divide both sides of the equation by 7. Therefore, \(x = 14/7 = 2\).
03

Check the Solution

Lastly, check the solution by substituting \(x = 2\) back into the original equation \(5x + 2x = 14\). After substitution, it becomes \(5*2 + 2*2 = 14\) or \(14 = 14\), which is correct. Therefore, the value of \(x = 2\) is the solution.

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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 important to combine like terms to simplify the equation. Like terms are terms whose variables (and their exponents) are the same. In the example exercise, we have terms with the variable 'x' being combined.
To combine like terms, you take the coefficient (the number in front of the variable) of each term and perform the necessary mathematical operations. In this case, we're adding the coefficients of 'x'.
  • Identify like terms: In the equation \(5x + 2x = 14\), both '5x' and '2x' are like terms because the variable 'x' is the same in both.
  • Add the coefficients: You simply add 5 and 2 to get 7.
  • Write the simplified equation: After combining like terms, you have \(7x = 14\).
A common mistake is to combine terms that are not like. Ensure that only the coefficients of like terms are combined, disregarding terms that have different variables or exponents.
Solving for Variables
Once you've combined like terms, the next step is to isolate the variable to solve the equation. The goal is to get the variable on one side of the equation, with a coefficient of 1, which represents its value.
  • Isolation: For the equation \(7x = 14\), the variable 'x' needs to be isolated. This is done by undoing whatever operation is being done to 'x'. Since '7x' means 7 multiplied by 'x', you'll do the opposite operation 鈥� division.
  • Dividing both sides: Divide both sides of the equation by 7 (the coefficient of 'x') to get \(x = 14/7\).
  • Simplify: After performing the division, the equation simplifies to \(x = 2\), which represents the solution.
It's crucial to perform any operation you do to one side of the equation to the other side as well, to maintain the equality.
Equation Checking
After finding a solution to an equation, it is a good practice to check if the solution is correct. This verification step ensures that the variable's value satisfies the original equation, maintaining the balance on both sides.
  • Substitute the solution: Take the solution \(x = 2\) and substitute it back into the original equation, replacing 'x' with 2.
  • Perform the operations: With substitution, the equation becomes \(5(2) + 2(2) = 14\), which simplifies to \(10 + 4 = 14\).
  • Verify equality: Confirm that both sides of the equation are equal (14 = 14) after the substitution, thus ensuring the solution is correct.
If the check fails (the sides aren't equal), it means there might be a mistake in the solution process, and you'll need to go back and review your steps.

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