Problem 3 - College Algebra

Question

solve and check each linear equation. $$ 11 x-(6 x-5)-40 $$

Step-by-Step Solution

Answer

The solution to the linear equation is \( x = 7 \)

Step 1: Simplify the equation

First, distribute the negative sign in the bracket. The equation then becomes: \(11x - 6x + 5 - 40\).

Step 2: Combine like terms

Following the rules of combining like terms, the equation simplifies to: \(5x - 35\)

Step 3: Isolate the variable x

To get x alone, add 35 to both sides of the equation. It becomes: \( 5x = 35 \). Now divide both sides by 5. The equation becomes: \( x = 7 \)

Step 4: Check the solution

Substitute \( x = 7 \) back into the original equation: \( 11 (7) - 6 (7) -5 - 40 = 0 \), thus verifying that the solution is correct

Key Concepts

Equation Simplification

Equation simplification is a fundamental process in solving linear equations. The goal is to make the equation as easy as possible to work with by reducing complexity. One common method is to eliminate parentheses by distributing any numbers or signs outside of them. In the case of the exercise, we need to distribute the negative sign to the terms inside the bracket:

• Distribute the negative sign: \(11x - (6x - 5) - 40\) becomes \(11x - 6x + 5 - 40\).

It's important to carry out this step carefully because a small mistake can change the entire solution. Distributing signs correctly sets the stage for the next steps in solving any linear equation. Take your time in this step to ensure accuracy.

Combining Like Terms

Combining like terms is a crucial step in simplifying equations further after distribution. Like terms are terms that contain the same variable raised to the same power. These terms can be added or subtracted from each other to simplify the equation.In the exercise, we see:

• Terms:
  • \(11x\)
  • \(-6x\)
• Constant:
  • \(+ 5\)
  • \(- 40\)

To combine these, add the coefficients of like terms together:

• \(11x - 6x = 5x\)
• \(+ 5 - 40 = -35\)

This simplifies our equation to \(5x - 35\). Combining like terms reduces the number of terms in the equation, making it easier to solve. Always double-check to ensure that all terms were combined correctly.

Variable Isolation

Variable isolation involves manipulating the equation so that the variable stands alone on one side. The ultimate goal here is to find the value of the variable.Once our equation is simplified to \(5x - 35\), the next step is to isolate \(x\):

• Add \(35\) to both sides to eliminate the constant term from the side with the variable:
  • \(5x - 35 + 35 = 35\)
  • Which simplifies to \(5x = 35\)
• Divide both sides by \(5\) to find \(x\):
  • \(\frac{5x}{5} = \frac{35}{5}\)
  • This results in \(x = 7\)

Clear and precise steps ensure you correctly isolate the variable. Keeping your work neat can help you avoid errors in this critical stage.

Solution Verification

Solution verification is the process of checking your work to ensure the solution is accurate. It's a crucial final step to confirm that the variable's value satisfies the original equation.In our exercise, the found value for \(x\) was \(7\). Plugging it back into the original equation, we check:

• Substitute \(x = 7\): \(11(7) - 6(7) + 5 - 40\)
• Calculate each part:
  • \(11 \times 7 = 77\)
  • \(6 \times 7 = 42\)
  • So the expression simplifies to: \(77 - 42 + 5 - 40\)
  • Which further simplifies to \(0\)

Since both sides of the equation match after substitution, the solution \(x = 7\) is verified as correct. Verifying ensures you can be confident in the accuracy of your solution. It's a good habit to always double-check your answers.