Solving X + Y = 10 And X = 4y: A Step-by-Step Guide

by Axel Sørensen 52 views

Hey guys! Let's dive into solving this system of equations. We have two equations here:

  1. x + y = 10
  2. x = 4y

We're going to find the values of x and y that satisfy both of these equations simultaneously. There are a couple of methods we can use, but let's go with the substitution method first because it looks pretty straightforward in this case. We can also verify the solution using the elimination method to ensure our answer is correct.

Step-by-Step Solution Using the Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This way, we reduce the problem to a single equation with a single variable, which is much easier to solve. Let's break it down:

Step 1: Solve one equation for one variable

Looking at our equations, the second one, x = 4y, is already solved for x. This is super convenient! We know that x is equal to 4y. This is a crucial piece of information because it allows us to directly substitute this expression into the first equation. It's like having a ready-made replacement part that fits perfectly into our puzzle.

Step 2: Substitute the expression into the other equation

Now, we'll take the expression 4y (which is equal to x) and plug it into the first equation, x + y = 10. This gives us:

(4y) + y = 10

Notice how we've replaced x with 4y. This substitution is the heart of this method. We've effectively eliminated x from the equation, leaving us with an equation that only involves y. This is a significant step because we now have a single equation with a single unknown, which is solvable.

Step 3: Solve for the remaining variable

We now have the equation 4y + y = 10. This is a simple algebraic equation that we can easily solve for y. Let's combine the like terms on the left side:

5y = 10

To isolate y, we'll divide both sides of the equation by 5:

y = 10 / 5

y = 2

Awesome! We've found the value of y. It's equal to 2. This is a major breakthrough because we now have one piece of the solution. Knowing the value of y makes it much easier to find the value of x.

Step 4: Substitute the value back to find the other variable

Now that we know y = 2, we can substitute this value back into either of the original equations to find x. The second equation, x = 4y, looks simpler, so let's use that one:

x = 4 * (2)

x = 8

Fantastic! We've found the value of x. It's equal to 8. We now have both values: x = 8 and y = 2. This is the solution to our system of equations.

Step 5: Check the solution

It's always a good idea to check our solution to make sure it's correct. We can do this by plugging the values of x and y back into both original equations:

  • Equation 1: x + y = 10*
    • 8 + 2 = 10* (This is true!)
  • Equation 2: x = 4y*
    • 8 = 4 * 2*
    • 8 = 8* (This is also true!)

Since our values satisfy both equations, we can be confident that our solution is correct.

Alternative Solution: Verification using the Elimination Method

To ensure the accuracy of our solution, let's use another method called the elimination method. This method involves manipulating the equations so that when we add or subtract them, one of the variables is eliminated. This gives us a single equation with a single variable, which we can then solve.

Step 1: Align the equations

First, we need to rewrite the equations so that the x and y terms are aligned:

  1. x + y = 10
  2. x - 4y = 0 (We subtracted 4y from both sides of the second original equation)

Step 2: Eliminate one variable

Notice that the coefficients of x in both equations are the same (1). This makes it easy to eliminate x. We can subtract the second equation from the first equation:

(x + y) - (x - 4y) = 10 - 0

x + y - x + 4y = 10

5y = 10

Step 3: Solve for the remaining variable

We're left with 5y = 10, which we already solved earlier: y = 2

Step 4: Substitute back to find the other variable

Now that we know y = 2, we can substitute it back into either of the original equations. Let's use the first equation, x + y = 10:

x + 2 = 10

x = 10 - 2

x = 8

Step 5: Check the solution (again!)

As before, we should check our solution. We already did this in the substitution method, and we know x = 8 and y = 2 satisfy both original equations.

Conclusion: The Solution

So, after using both the substitution method and verifying with the elimination method, we've confidently found the solution to the system of equations:

  • x = 8
  • y = 2

This means the point (8, 2) is the intersection of the two lines represented by these equations. Understanding how to solve systems of equations like this is super important in math and has lots of real-world applications in fields like engineering, economics, and computer science. Great job, guys! Keep practicing, and you'll become equation-solving pros in no time!