Certainly! Let's solve each pair of equations step-by-step to determine the correct matches.
### Pair 1:
[tex]\[ \begin{cases}
2x - 4y = 6 \\
-4x + 8y = -12
\end{cases} \][/tex]
1. Multiply the first equation by 2:
[tex]\[ 4x - 8y = 12 \][/tex]
2. Compare this with the second equation:
[tex]\[ -4x + 8y = -12 \][/tex]
3. Adding these two equations:
[tex]\[ (4x - 8y) + (-4x + 8y) = 12 + (-12) \][/tex]
[tex]\[ 0 = 0 \][/tex]
This implies that the equations are dependent (the same line), indicating an Infinite number of solutions.
### Pair 2:
[tex]\[ \begin{cases}
-2x + 6y = 4 \\
x - 3y = 2
\end{cases} \][/tex]
1. Solve the second equation for [tex]\( x \)[/tex]:
[tex]\[ x = 3y + 2 \][/tex]
2. Substitute [tex]\( x = 3y + 2 \)[/tex] into the first equation:
[tex]\[ -2(3y + 2) + 6y = 4 \][/tex]
[tex]\[ -6y - 4 + 6y = 4 \][/tex]
[tex]\[ -4 = 4 \][/tex]
This is a contradiction, so there is No solution.
### Pair 3:
[tex]\[ \begin{cases}
2x - 3y = 7 \\
-3x + 7y = -8
\end{cases} \][/tex]
1. We can solve this system using elimination or substitution. First, solve for one variable from one equation and substitute in the other. Let's multiply the first equation by 3 and the second by 2:
[tex]\[ \begin{cases}
6x - 9y = 21 \\
-6x + 14y = -16
\end{cases} \][/tex]
2. Adding these equations:
[tex]\[ (6x - 9y) + (-6x + 14y) = 21 - 16 \][/tex]
[tex]\[ 5y = 5 \][/tex]
[tex]\[ y = 1 \][/tex]
3. Substitute [tex]\( y = 1 \)[/tex] back into the first equation:
[tex]\[ 2x - 3(1) = 7 \][/tex]
[tex]\[ 2x - 3 = 7 \][/tex]
[tex]\[ 2x = 10 \][/tex]
[tex]\[ x = 5 \][/tex]
Thus, the solution is:
[tex]\[ (x, y) = (5, 1) \][/tex]
### Pair 4:
[tex]\[ \begin{cases}
4x - 3y = 9 \\
2x + 3y = 9
\end{cases} \][/tex]
1. Add the two equations together:
[tex]\[ (4x - 3y) + (2x + 3y) = 9 + 9 \][/tex]
[tex]\[ 6x = 18 \][/tex]
[tex]\[ x = 3 \][/tex]
2. Substitute [tex]\( x = 3 \)[/tex] back into the second equation:
[tex]\[ 2(3) + 3y = 9 \][/tex]
[tex]\[ 6 + 3y = 9 \][/tex]
[tex]\[ 3y = 3 \][/tex]
[tex]\[ y = 1 \][/tex]
Thus, the solution is:
[tex]\[ (x, y) = (3, 1) \][/tex]
Summary of our matches:
- Pair 1: Infinite number of solutions
- Pair 2: No solution
- Pair 3: (5, 1)
- Pair 4: (3, 1)
Therefore:
[tex]\[
\begin{array}{l}
2 x-4 y=6 \, \& \, -4 x+8 y=-12 \quad \text{match: Infinite number of solutions} \\
-2 x+6 y=4 \, \& \, x-3 y=2 \quad \text{match: No solution} \\
2 x-3 y=7 \, \& \, -3 x+7 y=-8 \quad \text{match: (5, 1)} \\
4 x-3 y=9 \, \& \, 2 x+3 y=9 \quad \text{match: (3, 1)}
\end{array}
\][/tex]
