Solving SPLDV By Substitution -x + Y = -1 And X + 2y = 7

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Hey guys! Ever wrestled with a system of linear equations? Don't worry, it's a common math hurdle, and we're here to help you jump over it. In this article, we're going to dive deep into solving a system of linear equations with two variables (SPLDV) using the substitution method. We'll take the specific example of -x + y = -1 and x + 2y = 7 and break it down step-by-step so you can conquer similar problems with confidence. So, buckle up and let's get started!

What is SPLDV?

Before we jump into the solution, let's quickly recap what SPLDV actually means. SPLDV stands for Sistem Persamaan Linear Dua Variabel, which translates to System of Linear Equations with Two Variables. Basically, it's a set of two or more linear equations that involve two variables, typically represented as x and y. The goal is to find the values of these variables that satisfy all equations simultaneously. Think of it as finding the point where two lines intersect on a graph. That point's coordinates (x, y) are the solution to the SPLDV.

Why Substitution Method?

There are several methods to solve SPLDV, including substitution, elimination, and graphing. The substitution method is particularly useful when one of the equations can be easily rearranged to express one variable in terms of the other. This makes it straightforward to substitute that expression into the other equation, reducing the problem to a single equation with a single variable. This method is like a clever trick that simplifies the problem, making it easier to crack. In our example, the equation -x + y = -1 can be easily rearranged to express y in terms of x, making substitution an ideal choice.

Step-by-Step Solution

Now, let's tackle our specific problem:

Equation 1: -x + y = -1 Equation 2: x + 2y = 7

Step 1: Isolate One Variable in One Equation

The first step in the substitution method is to isolate one variable in one of the equations. This means we want to get one variable by itself on one side of the equation. Looking at our equations, it seems easier to isolate 'y' in Equation 1 because it already has a coefficient of 1. So, let's rewrite Equation 1:

-x + y = -1

Add 'x' to both sides:

y = x - 1

Great! Now we have 'y' expressed in terms of 'x'. This is a crucial step as we'll use this expression in the next step.

Step 2: Substitute into the Other Equation

Now comes the fun part – substitution! We'll substitute the expression we found for 'y' (which is x - 1) into the other equation (Equation 2). This is where the magic happens, guys. We're essentially replacing 'y' in Equation 2 with its equivalent expression in terms of 'x'.

Equation 2: x + 2y = 7

Substitute y = x - 1:

x + 2(x - 1) = 7

See what we did there? We've now transformed Equation 2 into an equation with only one variable, 'x'. This is a huge step forward!

Step 3: Solve for the Remaining Variable

Now we have a single equation with a single variable. Time to solve for 'x'. Let's simplify and solve the equation:

x + 2(x - 1) = 7

Distribute the 2:

x + 2x - 2 = 7

Combine like terms:

3x - 2 = 7

Add 2 to both sides:

3x = 9

Divide both sides by 3:

x = 3

Awesome! We've found the value of 'x'. Now we know that x = 3. We're halfway there!

Step 4: Substitute Back to Find the Other Variable

We've found 'x', but we still need to find 'y'. This is where we substitute the value of 'x' back into either Equation 1 or Equation 2 (or even the rearranged equation y = x - 1) to solve for 'y'. It's usually easiest to substitute into the rearranged equation, so let's do that:

y = x - 1

Substitute x = 3:

y = 3 - 1

y = 2

Fantastic! We've found the value of 'y'. We now know that y = 2.

Step 5: Check Your Solution

Before we declare victory, it's always a good idea to check our solution. This ensures that the values we found for 'x' and 'y' actually satisfy both original equations. Let's plug x = 3 and y = 2 into Equation 1 and Equation 2:

Equation 1: -x + y = -1

Substitute x = 3 and y = 2:

-3 + 2 = -1

-1 = -1 (This is true!)

Equation 2: x + 2y = 7

Substitute x = 3 and y = 2:

3 + 2(2) = 7

3 + 4 = 7

7 = 7 (This is also true!)

Since our values for 'x' and 'y' satisfy both equations, we can confidently say that our solution is correct.

Solution

Therefore, the solution to the system of equations:

-x + y = -1 x + 2y = 7

is x = 3 and y = 2. We can also write this as an ordered pair: (3, 2).

Visualizing the Solution

To further understand what we've done, let's think about this graphically. Each linear equation represents a straight line on a graph. The solution to the SPLDV is the point where these two lines intersect. In our case, the lines represented by -x + y = -1 and x + 2y = 7 intersect at the point (3, 2). This point is the one and only solution that satisfies both equations.

Common Mistakes to Avoid

When solving SPLDV using substitution, there are a few common mistakes you should watch out for:

  • Forgetting to Distribute: When substituting an expression, make sure you distribute any coefficients correctly. For example, in our problem, we needed to distribute the 2 in x + 2(x - 1) = 7.
  • Substituting into the Same Equation: Don't substitute the expression you found back into the same equation you used to find it. This won't help you solve for the variables. You need to substitute into the other equation.
  • Sign Errors: Be extra careful with negative signs. A small sign error can throw off your entire solution.
  • Not Checking Your Solution: Always take the time to check your solution by plugging the values back into the original equations. This is the best way to catch any mistakes.

Practice Makes Perfect

The best way to master the substitution method is to practice, practice, practice! Try solving different SPLDV problems with varying levels of difficulty. The more you practice, the more comfortable and confident you'll become. You can find tons of practice problems in textbooks, online resources, and even worksheets.

Conclusion

So there you have it, guys! We've successfully solved the system of equations -x + y = -1 and x + 2y = 7 using the substitution method. We broke down the process into clear, easy-to-follow steps, and we even discussed common mistakes to avoid. Remember, the key to mastering SPLDV is understanding the concepts and practicing regularly. Keep up the great work, and you'll be solving these problems like a pro in no time!

If you have any questions or want to try another example, feel free to ask. Happy solving!