Solve these simultaneous equations by eliminating:
1) 7x - y = 9
2x + 3y = 19
2) x - 3y = 0
3x + y = 30
3) 7x - y = 19
2x + 3y = 12
Solve the following pair of linear equation by the elimination method and the substitution method 2 X + 3 y is equal to 0 and X + y is equal to 1
To solve these simultaneous equations by eliminating method, follow these steps:
1) 7x - y = 9
2x + 3y = 19
To eliminate the variables, we will multiply the first equation by 3 and the second equation by 1:
3(7x - y) = 3(9)
1(2x + 3y) = 1(19)
Which simplifies to:
21x - 3y = 27
2x + 3y = 19
Now, we can add the two equations together:
(21x - 3y) + (2x + 3y) = 27 + 19
This gives us:
23x = 46
Divide both sides by 23:
x = 2
Substitute the value of x back into one of the original equations, for example, the first equation:
7x - y = 9
7(2) - y = 9
14 - y = 9
Subtract 14 from both sides:
-y = 9 - 14
-y = -5
Multiply both sides by -1 to isolate y:
y = 5
So the solution to the simultaneous equations is x = 2 and y = 5.
2) x - 3y = 0
3x + y = 30
To eliminate the variables, we will multiply the first equation by 3 and the second equation by 1:
3(x - 3y) = 3(0)
1(3x + y) = 1(30)
Which simplifies to:
3x - 9y = 0
3x + y = 30
Now, we can subtract the second equation from the first:
(3x - 9y) - (3x + y) = 0 - 30
This gives us:
-10y = -30
Divide both sides by -10:
y = 3
Substitute the value of y back into one of the original equations, for example, the first equation:
x - 3y = 0
x - 3(3) = 0
x - 9 = 0
Add 9 to both sides:
x = 9
So the solution to the simultaneous equations is x = 9 and y = 3.
3) 7x - y = 19
2x + 3y = 12
To eliminate the variables, we will multiply the first equation by 3 and the second equation by 1:
3(7x - y) = 3(19)
1(2x + 3y) = 1(12)
Which simplifies to:
21x - 3y = 57
2x + 3y = 12
Now, we can add the two equations together:
(21x - 3y) + (2x + 3y) = 57 + 12
This gives us:
23x = 69
Divide both sides by 23:
x = 3
Substitute the value of x back into one of the original equations, for example, the first equation:
7x - y = 19
7(3) - y = 19
21 - y = 19
Subtract 21 from both sides:
-y = 19 - 21
-y = -2
Multiply both sides by -1 to isolate y:
y = 2
So the solution to the simultaneous equations is x = 3 and y = 2.
To solve simultaneous equations by eliminating, we need to get rid of one of the variables by manipulating the equations. Here's how to solve each of the given sets of simultaneous equations:
1) Start with the given equations:
Equation 1: 7x - y = 9
Equation 2: 2x + 3y = 19
To eliminate one variable, we can multiply Equation 1 by 3 and Equation 2 by 7:
Equation 3: 21x - 3y = 27
Equation 4: 14x + 21y = 133
Now, add Equation 3 and Equation 4 together:
(21x - 3y) + (14x + 21y) = 27 + 133
35x + 18y = 160
We have eliminated the variable "y". Let's call this new equation Equation 5.
To get rid of "y" completely, we need to eliminate it from both sides of Equation 5. Multiply Equation 1 by 18:
Equation 6: 126x - 18y = 162
Now, add Equation 5 and Equation 6 together:
(35x + 18y) + (126x - 18y) = 160 + 162
161x = 322
Finally, divide both sides of the equation by 161:
x = 322/161
x = 2
Substitute the value of x back into either Equation 1 or Equation 2. Let's use Equation 1:
7(2) - y = 9
14 - y = 9
-y = 9 - 14
-y = -5
y = 5
Hence, the solution for the simultaneous equations is x = 2 and y = 5.
2) Start with the given equations:
Equation 1: x - 3y = 0
Equation 2: 3x + y = 30
To eliminate one variable, we can multiply Equation 1 by 3 and Equation 2 by -1:
Equation 3: 3x - 9y = 0
Equation 4: -3x - y = -30
Now, add Equation 3 and Equation 4 together:
(3x - 9y) + (-3x - y) = 0 + (-30)
-10y = -30
Divide both sides of the equation by -10:
y = -30 / -10
y = 3
Substitute the value of y back into either Equation 1 or Equation 2. Let's use Equation 1:
x - 3(3) = 0
x - 9 = 0
x = 9
Therefore, the solution for the simultaneous equations is x = 9 and y = 3.
3) Start with the given equations:
Equation 1: 7x - y = 19
Equation 2: 2x + 3y = 12
To eliminate one variable, we can multiply Equation 1 by 3 and Equation 2 by 1:
Equation 3: 21x - 3y = 57
Equation 4: 2x + 3y = 12
Now, add Equation 3 and Equation 4 together:
(21x - 3y) + (2x + 3y) = 57 + 12
23x = 69
Divide both sides of the equation by 23:
x = 69 / 23
x = 3
Substitute the value of x back into either Equation 1 or Equation 2. Let's use Equation 1:
7(3) - y = 19
21 - y = 19
-y = 19 - 21
-y = -2
y = 2
Hence, the solution for the simultaneous equations is x = 3 and y = 2.
I'll do one; you try the others, using the same method.
7x - y = 9
2x + 3y = 19
to eliminate y, multiply the 1st by 3 to get
21x - 3y = 27
2x + 3y = 19
Now if you add the equations, the y terms are eliminated, giving
23x = 46
x = 2
so, y = 5
The others work the same way.