Sure! Let's solve each pair of simultaneous equations step by step.
### Pair 1:
[tex]\[ 3x - 2y + 3 = 0 \][/tex]
[tex]\[ 4x + 3y - 47 = 0 \][/tex]
1. First, rewrite the equations without constants for clarity:
[tex]\[ 3x - 2y = -3 \][/tex]
[tex]\[ 4x + 3y = 47 \][/tex]
2. Solve the above system of linear equations.
The solution to the system is:
[tex]\[ x = 5 \][/tex]
[tex]\[ y = 9 \][/tex]
### Pair 2:
[tex]\[ 4x - 5y - 16 = 0 \][/tex]
[tex]\[ 7x - 13y + 10 = 0 \][/tex]
1. Rewrite the equations:
[tex]\[ 4x - 5y = 16 \][/tex]
[tex]\[ 7x - 13y = -10 \][/tex]
2. Solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
The solution to the system is:
[tex]\[ x = -\frac{258}{17} \][/tex]
[tex]\[ y = -\frac{152}{17} \][/tex]
### Pair 3:
[tex]\[ 3x + 2y + 25 = 0 \][/tex]
[tex]\[ 2x + y + 10 = 0 \][/tex]
1. Rewrite the equations:
[tex]\[ 3x + 2y = -25 \][/tex]
[tex]\[ 2x + y = -10 \][/tex]
2. Solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
The solution to the system is:
[tex]\[ x = 5 \][/tex]
[tex]\[ y = -20 \][/tex]
### Pair 4:
[tex]\[ 2x + y = 35 \][/tex]
[tex]\[ 3x + 4y = 65 \][/tex]
1. The equations are already in standard form:
[tex]\[ 2x + y = 35 \][/tex]
[tex]\[ 3x + 4y = 65 \][/tex]
2. Solve for [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
The solution to the system is:
[tex]\[ x = 15 \][/tex]
[tex]\[ y = 5 \][/tex]
Hence, the solutions to the given pairs of equations are:
1. For [tex]\(3x - 2y + 3 = 0\)[/tex] and [tex]\(4x + 3y - 47 = 0\)[/tex]:
[tex]\[ x = 5, \quad y = 9 \][/tex]
2. For [tex]\(4x - 5y - 16 = 0\)[/tex] and [tex]\(7x - 13y + 10 = 0\)[/tex]:
[tex]\[ x = -\frac{258}{17}, \quad y = -\frac{152}{17} \][/tex]
3. For [tex]\(3x + 2y + 25 = 0\)[/tex] and [tex]\(2x + y + 10 = 0\)[/tex]:
[tex]\[ x = 5, \quad y = -20 \][/tex]
4. For [tex]\(2x + y = 35\)[/tex] and [tex]\(3x + 4y = 65\)[/tex]:
[tex]\[ x = 15, \quad y = 5 \][/tex]
