Which ordered pair is the solution to the system of equations below?
{3x-7y=-10
{5x-y=-6
A system of equations is shown below
2x+y=7
3x-2y=0
What is the solution to the system?
3x-7y=-10
5x-y=-6 ---> y = 5x+6
sub that into the first
3x - 7(5x+6) = -10
3x - 35x - 42 = -10
-32x = 32
x = -1
then y = 5(-1)+6 = 1
the ordered pair (-1,1) is the solution
do the same thing for the second problem
To find the solution to a system of equations, you need to solve the equations simultaneously by using a method such as substitution or elimination. Let's solve the given systems of equations step-by-step.
System of Equations 1:
3x - 7y = -10 ----(Equation 1)
5x - y = -6 ----(Equation 2)
Let's use the elimination method to solve this system:
1. Multiply Equation 2 by 7 to make the coefficients of 'y' in both equations opposite:
7(5x - y) = 7(-6)
35x - 7y = -42 ----(Equation 3)
2. Now, add Equation 1 and Equation 3 together to eliminate the 'y' variable:
(3x - 7y) + (35x - 7y) = (-10) + (-42)
38x - 14y = -52 ----(Equation 4)
3. Divide Equation 4 by 2 to simplify the equation:
(38x - 14y)/2 = -52/2
19x - 7y = -26 ----(Equation 5)
4. Next, multiply Equation 2 by 7 to make the coefficients of 'y' in both equations opposite:
7(3x - 2y) = 7(0)
21x - 14y = 0 ----(Equation 6)
5. Add Equation 5 and Equation 6 together to eliminate the 'y' variable:
(19x - 7y) + (21x - 14y) = (-26) + 0
40x - 21y = -26 ----(Equation 7)
6. Divide Equation 7 by 3 to simplify the equation:
(40x - 21y)/3 = -26/3
13x - 7y = -26/3 ----(Equation 8)
7. Now, we have two equations:
19x - 7y = -26 ----(Equation 5)
13x - 7y = -26/3 ----(Equation 8)
8. Subtract Equation 8 from Equation 5 to eliminate the 'x' variable:
(19x - 7y) - (13x - 7y) = (-26) - (-26/3)
19x - 13x = -26 + 26/3
6x = (-78 + 26)/3
6x = -52/3
9. Divide both sides by 6 to isolate 'x':
x = (-52/3)/6
x = -52/18
10. Simplify 'x':
x = -26/9
Now substitute the value of 'x' back into one of the original equations to find the value of 'y'.
Let's use Equation 2:
5x - y = -6
Substituting x = -26/9 into this equation:
5(-26/9) - y = -6
-130/9 - y = -6
To get 'y' alone, subtract -130/9 from both sides:
- y = -6 + 130/9
Combining the terms on the right side:
- y = (-54 + 130)/9
- y = 76/9
Therefore, the solution to the first system of equations is:
x = -26/9 and y = 76/9
So, the ordered pair (x, y) is (-26/9, 76/9).
System of Equations 2:
2x + y = 7 ----(Equation 1)
3x - 2y = 0 ----(Equation 2)
Let's solve this system using the substitution method:
1. Solve Equation 1 for 'y':
y = 7 - 2x
2. Substitute the expression for 'y' in terms of 'x' into Equation 2:
3x - 2(7 - 2x) = 0
3. Simplify and solve for 'x':
3x - 14 + 4x = 0
7x - 14 = 0
7x = 14
x = 14/7
x = 2
4. Substitute the value of 'x' back into Equation 1 to find 'y':
2(2) + y = 7
4 + y = 7
y = 7 - 4
y = 3
Therefore, the solution to the second system of equations is:
x = 2 and y = 3
So, the ordered pair (x, y) is (2, 3).
In conclusion,
The solution to the first system of equations is (-26/9, 76/9) and
the solution to the second system of equations is (2, 3).