To solve the system of equations using the elimination method, we start with the equations:
- \(-x + 4y = 7\)
- \(-3x + 7y = 1\)
First, we can manipulate the first equation to make the coefficients of \(x\) easier to align. We can multiply the entire first equation by 3 to eliminate \(x\) when we combine it with the second equation:
\[ 3(-x + 4y) = 3(7) \]
This gives us:
\[ -3x + 12y = 21 \quad \text{(Equation 3)} \]
Now our system of equations looks like this:
- \(-3x + 12y = 21\)
- \(-3x + 7y = 1\)
Next, we will subtract Equation 4 from Equation 3 to eliminate \(x\):
\[ (-3x + 12y) - (-3x + 7y) = 21 - 1 \]
Simplifying this gives:
\[ 12y - 7y = 20 \]
\[ 5y = 20 \]
Now, divide both sides by 5:
\[ y = 4 \]
Now that we have \(y\), we can substitute it back into one of the original equations to find \(x\). We'll use the first equation:
\[ -x + 4(4) = 7 \]
This simplifies to:
\[ -x + 16 = 7 \]
Subtract 16 from both sides:
\[ -x = 7 - 16 \]
\[ -x = -9 \]
Multiplying both sides by -1 gives:
\[ x = 9 \]
Thus, the solution to the system of equations is:
\[ \boxed{(9, 4)} \]
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