Solving Systems of Equations: Any Method | Interactive Worksheet ... - Free Printable

1. \( 3x + y = -8 \)
\( -4x + y = 6 \)
Subtract the second equation from the first:
\( (3x + y) - (-4x + y) = -8 - 6 \)
\( 3x + y + 4x - y = -14 \)
\( 7x = -14 \)
\( x = -2 \)
Substitute into first equation:
\( 3(-2) + y = -8 \)
\( -6 + y = -8 \)
\( y = -2 \)
Solution: \( (-2, -2) \)

2. \( y = -2x - 4 \)
\( y = 4x + 2 \)
Set equal:
\( -2x - 4 = 4x + 2 \)
\( -6x = 6 \)
\( x = -1 \)
Substitute into first equation:
\( y = -2(-1) - 4 = 2 - 4 = -2 \)
Solution: \( (-1, -2) \)

3. \( 4x - 5y = 6 \)
\( 2x + 5y = 18 \)
Add equations:
\( 6x = 24 \)
\( x = 4 \)
Substitute into first equation:
\( 4(4) - 5y = 6 \)
\( 16 - 5y = 6 \)
\( -5y = -10 \)
\( y = 2 \)
Solution: \( (4, 2) \)

4. \( x + 4y = -2 \)
\( 3x + 2y = 6 \)
Multiply first equation by 3:
\( 3x + 12y = -6 \)
Subtract second equation:
\( (3x + 12y) - (3x + 2y) = -6 - 6 \)
\( 10y = -12 \)
\( y = -1.2 \)
Substitute into first equation:
\( x + 4(-1.2) = -2 \)
\( x - 4.8 = -2 \)
\( x = 2.8 \)
Solution: \( (2.8, -1.2) \)

5. \( 2x + 3y = 9 \)
\( x = y - 2 \)
Substitute:
\( 2(y - 2) + 3y = 9 \)
\( 2y - 4 + 3y = 9 \)
\( 5y = 13 \)
\( y = 2.6 \)
\( x = 2.6 - 2 = 0.6 \)
Solution: \( (0.6, 2.6) \)

6. \( y = \frac{1}{3}x + 5 \)
\( y = 3x - 5 \)
Set equal:
\( \frac{1}{3}x + 5 = 3x - 5 \)
\( 5 + 5 = 3x - \frac{1}{3}x \)
\( 10 = \frac{8}{3}x \)
\( x = 10 \cdot \frac{3}{8} = 3.75 \)
\( y = 3(3.75) - 5 = 11.25 - 5 = 6.25 \)
Solution: \( (3.75, 6.25) \)

7. \( 5x + 2y = 14 \)
\( 3x + 4y = -14 \)
Multiply first by 2:
\( 10x + 4y = 28 \)
Subtract second:
\( (10x + 4y) - (3x + 4y) = 28 - (-14) \)
\( 7x = 42 \)
\( x = 6 \)
Substitute into first:
\( 5(6) + 2y = 14 \)
\( 30 + 2y = 14 \)
\( 2y = -16 \)
\( y = -8 \)
Solution: \( (6, -8) \)

8. \( x + 5y = 15 \)
\( 4x - 2y = 12 \)
Multiply first by 4:
\( 4x + 20y = 60 \)
Subtract second:
\( (4x + 20y) - (4x - 2y) = 60 - 12 \)
\( 22y = 48 \)
\( y = \frac{48}{22} = \frac{24}{11} \approx 2.18 \)
Substitute into first:
\( x + 5(\frac{24}{11}) = 15 \)
\( x + \frac{120}{11} = 15 \)
\( x = 15 - \frac{120}{11} = \frac{165 - 120}{11} = \frac{45}{11} \approx 4.09 \)
Solution: \( (\frac{45}{11}, \frac{24}{11}) \)

9. \( 5x - 9y = 9 \)
\( -4x + 6y = -4 \)
Multiply first by 2:
\( 10x - 18y = 18 \)
Multiply second by 3:
\( -12x + 18y = -12 \)
Add:
\( -2x = 6 \)
\( x = -3 \)
Substitute into first:
\( 5(-3) - 9y = 9 \)
\( -15 - 9y = 9 \)
\( -9y = 24 \)
\( y = -\frac{8}{3} \)
Solution: \( (-3, -\frac{8}{3}) \)

10. \( y = -x + 12 \)
\( y = x + 2 \)
Set equal:
\( -x + 12 = x + 2 \)
\( 12 - 2 = 2x \)
\( 10 = 2x \)
\( x = 5 \)
\( y = 5 + 2 = 7 \)
Solution: \( (5, 7) \)

11. \( y = -2x + 1 \)
\( 3x + 2y = 5 \)
Substitute:
\( 3x + 2(-2x + 1) = 5 \)
\( 3x - 4x + 2 = 5 \)
\( -x = 3 \)
\( x = -3 \)
\( y = -2(-3) + 1 = 6 + 1 = 7 \)
Solution: \( (-3, 7) \)

12. \( 4x - 5y = 12 \)
\( 3x + 2y = 32 \)
Multiply first by 2:
\( 8x - 10y = 24 \)
Multiply second by 5:
\( 15x + 10y = 160 \)
Add:
\( 23x = 184 \)
\( x = 8 \)
Substitute into first:
\( 4(8) - 5y = 12 \)
\( 32 - 5y = 12 \)
\( -5y = -20 \)
\( y = 4 \)
Solution: \( (8, 4) \)