Solve 13-16s = 15s-1

This algebraic equation might look a little intimidating at first glance, but breaking it down step-by-step makes it quite manageable. Our goal is to isolate the variable '$s$' on one side of the equation to find its value. We'll achieve this by using inverse operations, moving terms around the equals sign while maintaining the balance of the equation. Let's dive into the process of solving $13 - 16s = 15s - 1$.

Step 1: Combine 's' terms

The first logical step in solving for '$s$' is to gather all terms containing '$s$' onto one side of the equation. Currently, we have '-16s' on the left side and '15s' on the right. To bring them together, we can perform an inverse operation. Let's choose to add '16s' to both sides of the equation. This will eliminate the '-16s' term from the left side. Remember, whatever we do to one side, we must do to the other to keep the equation true.

So, we start with: $13 - 16s = 15s - 1$

Add '16s' to both sides: $13 - 16s + 16s = 15s - 1 + 16s$

This simplifies to: $13 = 31s - 1$

Now, all our '$s$' terms are on the right side of the equation. This is a good start, and we're one step closer to isolating '$s$'.

Step 2: Combine constant terms

With the '$s$' terms consolidated, the next step is to gather all the constant terms (numbers without variables) onto the opposite side of the equation. Currently, we have '13' on the left and '-1' on the right, alongside the '31s'. To move the '-1' to the left side, we perform the inverse operation: add '1' to both sides of the equation.

We have: $13 = 31s - 1$

Add '1' to both sides: $13 + 1 = 31s - 1 + 1$

This simplifies to: $14 = 31s$

At this point, we have a constant on the left and the '$s$' term on the right. The '$s$' variable is very close to being isolated.

Step 3: Isolate 's'

The final step to solve for '$s$' is to isolate it completely. We currently have '$31s$' on the right side, which means '31 multiplied by s'. To undo this multiplication, we perform the inverse operation: division. We need to divide both sides of the equation by '31'.

Our equation is: $14 = 31s$

Divide both sides by '31': rac{14}{31} = rac{31s}{31}

This gives us the solution: s = rac{14}{31}

So, the value of '$s$' that satisfies the equation $13 - 16s = 15s - 1$ is rac{14}{31}.

Step 4: Verification (Optional but Recommended)

To ensure our answer is correct, it's always a good idea to substitute the value of '$s$' back into the original equation and check if both sides are equal. This verification step helps catch any potential errors made during the solving process.

Original equation: $13 - 16s = 15s - 1$

Substitute s = rac{14}{31}: 13 - 16ig(rac{14}{31}ig) = 15ig(rac{14}{31}ig) - 1

Calculate the left side: 13 - rac{16 imes 14}{31} = 13 - rac{224}{31} To subtract, we need a common denominator: rac{13 imes 31}{31} - rac{224}{31} = rac{403}{31} - rac{224}{31} = rac{403 - 224}{31} = rac{179}{31}

Calculate the right side: 15ig(rac{14}{31}ig) - 1 = rac{15 imes 14}{31} - 1 = rac{210}{31} - 1 To subtract, we need a common denominator: rac{210}{31} - rac{1 imes 31}{31} = rac{210}{31} - rac{31}{31} = rac{210 - 31}{31} = rac{179}{31}

Since the left side (rac{179}{31}) equals the right side (rac{179}{31}), our solution s = rac{14}{31} is indeed correct.

Conclusion

Solving linear equations like $13 - 16s = 15s - 1$ involves a systematic approach of using inverse operations to isolate the variable. By combining like terms and carefully applying addition, subtraction, multiplication, and division to both sides of the equation, we successfully found that s = rac{14}{31}. This process is fundamental in algebra and is applicable to a wide range of mathematical problems. For further practice with algebraic equations, you can explore resources on Khan Academy or look into the principles of solving linear equations.