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Chapter 8: Problem 195

Solve the linear equation using the general strategy. $$5+6(3 s-5)=-3+2(8 s-1)$$

Short Answer

Expert verified
s = 10

Step by step solution

01

- Expand the brackets

First, expand the brackets on both sides of the equation. The left side: \[5 + 6(3s - 5) = 5 + 18s - 30 = 18s - 25\] The right side: \[-3 + 2(8s - 1) = -3 + 16s - 2 = 16s - 5\]
02

- Write the simplified equation

After expanding, the equation becomes: \[18s - 25 = 16s - 5\]
03

- Collect like terms on one side

Move the terms involving \(s\) to one side and the constant terms to the other side of the equation: \[18s - 16s = -5 + 25\] Combining the like terms gives: \[2s = 20\]
04

- Solve for \(s\)

Divide both sides of the equation by 2 to solve for \(s\): \[s = \frac{20}{2} = 10\]

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Key Concepts

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

Expanding Brackets
One of the first steps in solving a linear equation is expanding brackets. This means you need to distribute the numbers outside the brackets to every term inside the brackets. Let's look at an example from the given equation:

  • The left side: \(5 + 6(3s - 5)\) becomes \(5 + 18s - 30\)
  • The right side: \(-3 + 2(8s - 1)\) becomes \(-3 + 16s - 2\)
Now, simplify both sides by combining the constants and terms:

  • Left side: \(5 + 18s - 30\) simplifies to \(-25 + 18s\)
  • Right side: \(-3 + 16s - 2\) simplifies to \(-5 + 16s\)
Expanding brackets correctly is crucial as it sets up the equation for the next steps.
Combining Like Terms
After expanding brackets, the next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. This step simplifies the equation further ensuring all similar terms are grouped together.

For our example, after expanding the brackets the equation becomes:\( 18s - 25 = 16s - 5 \)

Here, you need to move all terms involving \( s \) to one side and constants to the other side. This will make it easier to solve for the variable. To do this, follow these steps:
  • Subtract \( 16s \) from both sides to keep the variable terms on the left: \( 18s - 16s - 25 = -5 \)
  • Simplify the equation: \( 2s - 25 = -5 \)
  • Next, add 25 to both sides to isolate the variable term: \( 2s = 20 \)
A key point here is to perform the same operation on both sides of the equation to maintain equality. Combining like terms correctly sets up the final step of isolating the variable.
Solving for a Variable
The final step in solving a linear equation is to isolate the variable. This means getting the variable by itself on one side of the equation. From our current simplified equation \( 2s = 20 \), this process involves isolating \( s \):

  • Divide both sides by 2: \( s = \frac{20}{2} = 10 \)
And there you go! You've found the value of \( s \). By ensuring every step - from expanding brackets, combining like terms, and solving for a variable - is followed precisely, you can confidently solve linear equations. Remember, practice makes perfect, so keep at it to improve your skills.

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