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 start with the equations:

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

We can substitute Equation 1 into Equation 2. First, let's rewrite Equation 2 using the expression for \( x \):

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

Now, let's simplify this equation:

1. Distribute \(-19\):

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

2. Combine like terms:

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

3. Isolate \( y \):

\[
-265y = -7 - 2603
\]
\[
-265y = -2610
\]
\[
y = \frac{-2610}{-265} = \frac{2610}{265} = 9.869 (approx)
\]

4. Now, calculate \( x \) using \( y = 9.869 \) in Equation 1:

\[
x = 14(9.869) - 137
\]
\[
x \approx 138.186 - 137
\]
\[
x \approx 1.186
\]

So, the approximate solutions are:

\[
(x, y) \approx (1.186, 9.869)
\]

In exact terms, the ordered pair of solutions (if derived algebraically from integer-coefficient equations) would not yield in simple integers. From numeric approaches or exact fractions, the values of \( (x,y) \) can be represented as \( (1, \frac{15}{3})\).

The final answer as an ordered pair is:

\[
\boxed{(1.186, 9.869)}
\]