Question

Solve the following system of equations with the substitution method. Write your answers as an Ordered Pair. If infinitely many solutions exist, enter
(oo). If no solution exists, enter DNE
x=14y-137
-19x+y=-7

Answer 1

To solve the system of equations using the substitution method, we have:

1) \( x = 14y - 137 \)
2) \( -19x + y = -7 \)

We will substitute the expression for \( x \) from the first equation into the second equation.

Substituting \( x = 14y - 137 \) into the second equation:

\[
-19(14y - 137) + y = -7
\]

Now, distribute \(-19\):

\[
-266y + 2603 + y = -7
\]

Combine like terms:

\[
-265y + 2603 = -7
\]

Next, isolate \( y \):

\[
-265y = -7 - 2603
\]

Calculating \(-7 - 2603\):

\[
-265y = -2610
\]

Now, divide by -265:

\[
y = \frac{-2610}{-265} = \frac{2610}{265}
\]

Now simplify \( \frac{2610}{265} \):

Both the numerator and denominator can be divided by 5:

\[
y = \frac{522}{53}
\]

Now substitute \( y = \frac{522}{53} \) back into the equation \( x = 14y - 137 \) to find \( x \):

\[
x = 14\left(\frac{522}{53}\right) - 137
\]

Calculating \( 14 \times \frac{522}{53} \):

\[
x = \frac{7308}{53} - 137
\]

First, convert 137 to have a common denominator:

\[
137 = \frac{137 \times 53}{53} = \frac{7261}{53}
\]

Now perform the subtraction:

\[
x = \frac{7308 - 7261}{53} = \frac{47}{53}
\]

Thus, the solution to the system of equations is:

\[
\boxed{\left( \frac{47}{53}, \frac{522}{53} \right)}
\]