Asked by Kawaiiskibidi
                Solve these equations with the elimination method.
X+2y=13
-x+y=5
            
        X+2y=13
-x+y=5
Answers
                    Answered by
            Kawaiiskibidi
            
    9x+y=2
-4x-y=-17
    
-4x-y=-17
                    Answered by
            GPT-4o mini
            
    To solve the system of equations using the elimination method, we start with the given equations:
1. \( x + 2y = 13 \) (Equation 1)
2. \( -x + y = 5 \) (Equation 2)
Our goal is to eliminate one of the variables by adding or subtracting the equations.
Let's first manipulate Equation 2 to have its \( x \) term in positive form, so we multiply it by -1:
\[
x - y = -5 \quad \text{(Equation 2 multiplied by -1)}
\]
Now we have:
1. \( x + 2y = 13 \)
2. \( x - y = -5 \)
Next, we can align these two equations:
\[
\begin{align*}
1: & \quad x + 2y = 13 \\
2: & \quad x - y = -5 \\
\end{align*}
\]
Now, we can eliminate \( x \) by subtracting Equation 2 from Equation 1:
\[
(x + 2y) - (x - y) = 13 - (-5)
\]
This simplifies to:
\[
x + 2y - x + y = 13 + 5
\]
\[
3y = 18
\]
Now, divide both sides by 3:
\[
y = 6
\]
Now that we have \( y \), we can substitute it back into one of the original equations to find \( x \). We'll use Equation 1:
\[
x + 2(6) = 13
\]
This simplifies to:
\[
x + 12 = 13
\]
Subtracting 12 from both sides gives:
\[
x = 1
\]
Thus, the solution to the system of equations is:
\[
x = 1, \quad y = 6
\]
So, the final solution is \( (1, 6) \).
    
1. \( x + 2y = 13 \) (Equation 1)
2. \( -x + y = 5 \) (Equation 2)
Our goal is to eliminate one of the variables by adding or subtracting the equations.
Let's first manipulate Equation 2 to have its \( x \) term in positive form, so we multiply it by -1:
\[
x - y = -5 \quad \text{(Equation 2 multiplied by -1)}
\]
Now we have:
1. \( x + 2y = 13 \)
2. \( x - y = -5 \)
Next, we can align these two equations:
\[
\begin{align*}
1: & \quad x + 2y = 13 \\
2: & \quad x - y = -5 \\
\end{align*}
\]
Now, we can eliminate \( x \) by subtracting Equation 2 from Equation 1:
\[
(x + 2y) - (x - y) = 13 - (-5)
\]
This simplifies to:
\[
x + 2y - x + y = 13 + 5
\]
\[
3y = 18
\]
Now, divide both sides by 3:
\[
y = 6
\]
Now that we have \( y \), we can substitute it back into one of the original equations to find \( x \). We'll use Equation 1:
\[
x + 2(6) = 13
\]
This simplifies to:
\[
x + 12 = 13
\]
Subtracting 12 from both sides gives:
\[
x = 1
\]
Thus, the solution to the system of equations is:
\[
x = 1, \quad y = 6
\]
So, the final solution is \( (1, 6) \).
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