Question

Solve using substitution or elimination. Tell whether the system has one, infinite, or no solutions.

1.) x-3y=-7, 2x=6y-14
2.) 9x + 8y =15, 9x+8y = 30
3.) 5x-9y = -43, 3x + 8y = 68

Answer 1

Let's solve the three systems of equations one by one.

### 1) System of Equations:
1. \( x - 3y = -7 \) (Equation 1)
2. \( 2x = 6y - 14 \) (Equation 2)

First, let's rearrange Equation 2:

\[
2x + 14 = 6y \implies y = \frac{2x + 14}{6} = \frac{x + 7}{3}
\]

Now, we can substitute \( y \) from this expression into Equation 1:

\[
x - 3\left(\frac{x + 7}{3}\right) = -7
\]

This simplifies to:

\[
x - (x + 7) = -7
\]
\[
x - x - 7 = -7 \implies -7 = -7
\]

This is a true statement, meaning the equations represent the same line.

**Conclusion**: The system has **infinite solutions**.

---

### 2) System of Equations:
1. \( 9x + 8y = 15 \) (Equation 1)
2. \( 9x + 8y = 30 \) (Equation 2)

Here, we can see that the left-hand sides of both equations are the same, but the right-hand sides are different (\(15\) and \(30\)).

**Conclusion**: The system has **no solutions** (the lines are parallel).

---

### 3) System of Equations:
1. \( 5x - 9y = -43 \) (Equation 1)
2. \( 3x + 8y = 68 \) (Equation 2)

We will use the elimination method for this system. We can eliminate \(y\) by manipulating the equations. Let's multiply Equation 1 by \(8\) and Equation 2 by \(9\) to align the coefficients of \(y\):

\[
8(5x - 9y) = 8(-43) \implies 40x - 72y = -344 \quad (Equation 3)
\]
\[
9(3x + 8y) = 9(68) \implies 27x + 72y = 612 \quad (Equation 4)
\]

Now, we add Equations 3 and 4 to eliminate \(y\):

\[
(40x - 72y) + (27x + 72y) = -344 + 612
\]
This simplifies to:

\[
67x = 268 \implies x = \frac{268}{67} = 4
\]

Now that we have \(x\), we can substitute it back to find \(y\). We can use either original equation; let’s use Equation 1:

\[
5(4) - 9y = -43
\]
\[
20 - 9y = -43 \implies -9y = -43 - 20
\]
\[
-9y = -63 \implies y = \frac{-63}{-9} = 7
\]

**Conclusion**: The system has **one solution** \( (x, y) = (4, 7) \).

---

### Summary of Results:
1. **System 1**: Infinite solutions.
2. **System 2**: No solutions.
3. **System 3**: One solution \((4, 7)\).