Question

Which system of equations has one solution?
–5x – 10y = 24
5x + 10y = 16
3x + 4y = –7
–3x – 4y = 7
–12x + 8y = 16
12x – 8y = 16
–2x + 7y = –5
2x + 7y = –9

Answer 1

To determine which system of equations has one solution, we need to analyze each pair of equations. A system of equations will have one solution if the equations are consistent and intersect at a single point, meaning they are not parallel or identical.

Let's analyze each of the given pairs:

1.
\[
-5x - 10y = 24 \quad (1)
\]
\[
5x + 10y = 16 \quad (2)
\]

If we add (1) and (2):
\[
0 = 40 \quad \text{(inconsistent)}
\]
This system has no solutions (they are parallel).

2.
\[
3x + 4y = -7 \quad (3)
\]
\[
-3x - 4y = 7 \quad (4)
\]

If we add (3) and (4):
\[
0 = 0 \quad \text{(consistent and identical)}
\]
This system has infinitely many solutions (they are identical).

3.
\[
-12x + 8y = 16 \quad (5)
\]
\[
12x - 8y = 16 \quad (6)
\]

Multiplying (5) by -1 yields:
\[
12x - 8y = -16 \quad (7)
\]
Comparing (6) and (7):
\[
12x - 8y = 16 \quad \text{(inconsistent)}
\]
This system has no solutions (they are parallel).

4.
\[
-2x + 7y = -5 \quad (8)
\]
\[
2x + 7y = -9 \quad (9)
\]

If we add (8) and (9):
\[
0 + 14y = -14 \implies y = -1 \quad \text{substituting back gives a unique solution for } x:
\]
\[
-2x + 7(-1) = -5 \implies -2x - 7 = -5 \implies -2x = 2 \implies x = -1
\]

Thus, the solution is \( (x, y) = (-1, -1) \).

The only system of equations with exactly one solution is:

\[
-2x + 7y = -5
\]
\[
2x + 7y = -9
\]

So, this pair of equations is the answer.