which method would be the simplest way to solve the system?
-3x + 67 = 10
-3x + 6y = -4
To solve this system of linear equations, you can use the substitution or elimination method. However, the simplest way to solve this system would likely be the elimination method, because the coefficient of \(x\) is the same in both equations. Here's the step-by-step process:
1. Arrange the equations to line up the variables:
\[
\begin{align*}
-3x + 0y &= 10 \\
-3x + 6y &= -4
\end{align*}
\]
2. Since the \(x\) terms are already lined up with the same coefficients but opposite signs, you can subtract the second equation from the first to eliminate \(x\):
\[
\begin{align*}
(-3x + 67) - (-3x + 6y) &= 10 - (-4) \\
-3x + 67 + 3x - 6y &= 10 + 4 \\
67 - 6y &= 14
\end{align*}
\]
3. Now, you're left with one variable to solve for:
\[
-6y = 14 - 67
\]
4. Simplify the equation:
\[
-6y = -53
\]
5. Solve for \(y\):
\[
y = \frac{-53}{-6} = \frac{53}{6}
\]
6. With the value of \(y\), you can now substitute it into one of the original equations to find \(x\):
\[
-3x + 6\left(\frac{53}{6}\right) = -4
\]
7. Simplify the equation:
\[
-3x + 53 = -4
\]
8. Move 53 to the other side of the equation:
\[
-3x = -4 - 53
\]
9. Simplify:
\[
-3x = -57
\]
10. Finally, solve for \(x\):
\[
x = \frac{-57}{-3} = 19
\]
So the solution to the system of equations is:
\[
x = 19, \quad y = \frac{53}{6}
\]