SIMULTANEOUS EQUATIONS WORKSHEET ANSWERS
ANSWERS:
**Problem 1:**
Equations:
1. 2x + y = 10
2. x - 3y = 1
Solution:
From equation 2, we can express x in terms of y: x = 3y + 1.
Substitute x in equation 1: 2(3y + 1) + y = 10.
Simplify: 6y + 2 + y = 10.
Combine like terms: 7y + 2 = 10.
Subtract 2 from both sides: 7y = 8.
Divide by 7: y = 8/7.
Substitute y back into x = 3y + 1: x = 3(8/7) + 1.
Simplify: x = 24/7 + 1.
Common denominator: x = (24 + 7)/7.
Simplify: x = 31/7.
So, the solution is x = 31/7 and y = 8/7.
**Problem 2:**
Equations:
1. 3x - 2y = 5
2. 4x + y = 7
Solution:
Multiply equation 2 by 2 to match the coefficient of y: 8x + 2y = 14.
Add equation 1 to the modified equation 2: 3x - 2y + 8x + 2y = 5 + 14.
Combine like terms: 11x = 19.
Divide by 11: x = 19/11.
Substitute x into equation 2: 4(19/11) + y = 7.
Simplify: 76/11 + y = 7.
Subtract 76/11 from both sides: y = 77/11 - 76/11.
Simplify: y = 1/11.
So, the solution is x = 19/11 and y = 1/11.
(Note: In this case, you can also use the elimination method.)
**Problem 3:**
Equations:
1. x + 2y = 4
, 2. 3x - y = 7
Solution:
Add equation 2 to equation 1 to eliminate y: x + 2y + 3x - y = 4 + 7.
Combine like terms: 4x + y = 11.
Solve the above equation for y: y = 11 - 4x.
Substitute y into equation 1: x + 2(11 - 4x) = 4.
Simplify: x + 22 - 8x = 4.
Combine like terms: -7x = -18.
Divide by -7: x = 18/7.
Substitute x back into y = 11 - 4x: y = 11 - 4(18/7).
Simplify: y = 77/7 - 72/7.
Simplify further: y = 5/7.
So, the solution is x = 18/7 and y = 5/7.
**Problem 4:**
Equations:
1. 2x - 5y = 8
2. 3x + y = 10
Solution:
Multiply equation 2 by 5 to match the coefficient of y: 15x + 5y = 50.
Add equation 1 to the modified equation 2: 2x - 5y + 15x + 5y = 8 + 50.
Combine like terms: 17x = 58.
Divide by 17: x = 58/17.
Substitute x into equation 2: 3(58/17) + y = 10.
Simplify: 174/17 + y = 10.
Subtract 174/17 from both sides: y = 170/17 - 174/17.
Simplify: y = -4/17.
So, the solution is x = 58/17 and y = -4/17.
(Note: In this case, you can also use the substitution method.)
**Problem 5:**
Equations:
1. 4x + 3y = 20
2. 2x - y = 7
Solution:
Multiply equation 2 by 3 to match the coefficient of y: 6x - 3y = 21.
Add equation 1 to the modified equation 2: 4x + 3y + 6x - 3y = 20 + 21.
Combine like terms: 10x = 41.
ANSWERS:
**Problem 1:**
Equations:
1. 2x + y = 10
2. x - 3y = 1
Solution:
From equation 2, we can express x in terms of y: x = 3y + 1.
Substitute x in equation 1: 2(3y + 1) + y = 10.
Simplify: 6y + 2 + y = 10.
Combine like terms: 7y + 2 = 10.
Subtract 2 from both sides: 7y = 8.
Divide by 7: y = 8/7.
Substitute y back into x = 3y + 1: x = 3(8/7) + 1.
Simplify: x = 24/7 + 1.
Common denominator: x = (24 + 7)/7.
Simplify: x = 31/7.
So, the solution is x = 31/7 and y = 8/7.
**Problem 2:**
Equations:
1. 3x - 2y = 5
2. 4x + y = 7
Solution:
Multiply equation 2 by 2 to match the coefficient of y: 8x + 2y = 14.
Add equation 1 to the modified equation 2: 3x - 2y + 8x + 2y = 5 + 14.
Combine like terms: 11x = 19.
Divide by 11: x = 19/11.
Substitute x into equation 2: 4(19/11) + y = 7.
Simplify: 76/11 + y = 7.
Subtract 76/11 from both sides: y = 77/11 - 76/11.
Simplify: y = 1/11.
So, the solution is x = 19/11 and y = 1/11.
(Note: In this case, you can also use the elimination method.)
**Problem 3:**
Equations:
1. x + 2y = 4
, 2. 3x - y = 7
Solution:
Add equation 2 to equation 1 to eliminate y: x + 2y + 3x - y = 4 + 7.
Combine like terms: 4x + y = 11.
Solve the above equation for y: y = 11 - 4x.
Substitute y into equation 1: x + 2(11 - 4x) = 4.
Simplify: x + 22 - 8x = 4.
Combine like terms: -7x = -18.
Divide by -7: x = 18/7.
Substitute x back into y = 11 - 4x: y = 11 - 4(18/7).
Simplify: y = 77/7 - 72/7.
Simplify further: y = 5/7.
So, the solution is x = 18/7 and y = 5/7.
**Problem 4:**
Equations:
1. 2x - 5y = 8
2. 3x + y = 10
Solution:
Multiply equation 2 by 5 to match the coefficient of y: 15x + 5y = 50.
Add equation 1 to the modified equation 2: 2x - 5y + 15x + 5y = 8 + 50.
Combine like terms: 17x = 58.
Divide by 17: x = 58/17.
Substitute x into equation 2: 3(58/17) + y = 10.
Simplify: 174/17 + y = 10.
Subtract 174/17 from both sides: y = 170/17 - 174/17.
Simplify: y = -4/17.
So, the solution is x = 58/17 and y = -4/17.
(Note: In this case, you can also use the substitution method.)
**Problem 5:**
Equations:
1. 4x + 3y = 20
2. 2x - y = 7
Solution:
Multiply equation 2 by 3 to match the coefficient of y: 6x - 3y = 21.
Add equation 1 to the modified equation 2: 4x + 3y + 6x - 3y = 20 + 21.
Combine like terms: 10x = 41.
