solve each system
1) 3x + y = 6
2x + 3y = 13
2) 2x + 5y - 6 = 0
-2x + y - 2 = 0
3) x + 3y = 6
4x - 5y = 7
I'll solve the first one, leaving you a chance to practice with the rest.
3x + y = 6 .....(1)
2x + 3y = 13 ....(2)
Multiply (1) by 3 and subtract (2) to eliminate y:
9x + 3y = 18 .....(1a)
2x + 3y = 13 .....(2a)
Subtract:
7x = 5
x=5/7 .....(3)
Substitute (3) in 1
3(5/7) + y = 6
y = 6-15/7
=27/7 ....(4)
Substitute (3) and (4) in (2) to check:
2(5/7) + 3(27/7)
=(10+81)/7
=13 OK.
To solve each system of equations, we can use either the substitution method or the elimination method. I'll explain both methods below.
1) 3x + y = 6
2x + 3y = 13
Substitution Method:
- Solve one of the equations for one variable in terms of the other variable.
From the first equation, solve for y:
y = 6 - 3x.
- Substitute the expression for y in the second equation with the y value from the first equation:
2x + 3(6 - 3x) = 13.
- Simplify and solve for x:
2x + 18 - 9x = 13,
-7x + 18 = 13,
-7x = 13 - 18,
-7x = -5,
x = -5 / -7,
x = 5/7.
- Substitute the x value back into one of the original equations to solve for y:
3(5/7) + y = 6,
15/7 + y = 42/7,
y = 42/7 - 15/7,
y = 27/7.
So the solution is x = 5/7 and y = 27/7.
Elimination Method:
- Multiply the first equation by 2 and the second equation by 3 in order to eliminate the x coefficient:
6x + 2y = 12,
6x + 9y = 39.
- Subtract the first equation from the second equation to eliminate x:
(6x + 9y) - (6x + 2y) = 39 - 12,
7y = 27,
y = 27 / 7.
- Substitute the y value into one of the original equations and solve for x:
3x + (27/7) = 6,
21x + 27 = 42,
21x = 42 - 27,
21x = 15,
x = 15 / 21,
x = 5/7.
The solution is x = 5/7 and y = 27/7.
2) 2x + 5y - 6 = 0
-2x + y - 2 = 0
Elimination Method:
- Add the two equations to eliminate the x variable:
(2x + 5y - 6) + (-2x + y - 2) = 0 + 0,
5y - 6 + y = 0 + 2,
6y - 6 = 2,
6y = 2 + 6,
6y = 8,
y = 8 / 6,
y = 4/3.
- Substitute the y value into one of the original equations and solve for x:
2x + 5(4/3) - 6 = 0,
2x + 20/3 - 18/3 = 0,
2x + 2/3 = 0,
2x = -2/3,
x = -2/3 / 2,
x = -1/3.
The solution is x = -1/3 and y = 4/3.
3) x + 3y = 6
4x - 5y = 7
Substitution Method:
- Solve one of the equations for one variable in terms of the other variable.
From the first equation, solve for x:
x = 6 - 3y.
- Substitute the expression for x in the second equation with the x value from the first equation:
4(6 - 3y) - 5y = 7.
- Simplify and solve for y:
24 - 12y - 5y = 7,
-17y = 7 - 24,
-17y = -17,
y = -17 / -17,
y = 1.
- Substitute the y value back into one of the original equations to solve for x:
x + 3(1) = 6,
x + 3 = 6,
x = 6 - 3,
x = 3.
So the solution is x = 3 and y = 1.
Elimination Method:
- Multiply the first equation by 4 and the second equation by 1 in order to eliminate the y coefficient:
4x + 12y = 24,
4x - 5y = 7.
- Subtract the second equation from the first equation to eliminate y:
(4x + 12y) - (4x - 5y) = 24 - 7,
4x + 12y - 4x + 5y = 17,
17y = 17,
y = 17 / 17,
y = 1.
- Substitute the y value into one of the original equations and solve for x:
x + 3(1) = 6,
x + 3 = 6,
x = 6 - 3,
x = 3.
The solution is x = 3 and y = 1.