Solve the elimination method
7a + 6b = 3
-7a +b = 25
The solution of the system is?
7a + 6b = 3
-7a + b = 25
b = 7a + 25
Solve for a
7a + 6(7a + 25) = 3
7a + 42a + 150 = 3
49a + 150 = 3
49a = -147
a = -3
Solve for b
-7(-3) + b = 25
21 + b = 25
b = 4
Check answers
7a + 6b = 3
7(-3) + 6(4) = 3
-21 + 24 = 3
-7a + b = 25
-7(-3) + 4 = 25
-21 + 4 = 25
Answers:
a = -3
b = 4
7 a + 6 b = 3
+
-7 a + b = 25
________________
7 a + ( - 7 a ) + 6 b + b = 3 + 25
7 b = 28 Divide both sides by 7
7 b / 7 = 28 / 7
b = 4
7 a + 6 b = 3
7 a + 6 * 4 = 3
7 a + 24 = 3 Subtract 24 to both sides
7 a + 24 - 24 = 3 - 24
7 a = - 21 Divide both sides by 7
7 a / 7 = - 21 / 7
a = - 3
Proof :
-7 a + b = 25
- 7 * ( - 3 ) + 4 = 25
21 + 4 = 25
To solve this system of equations using the elimination method, we can eliminate the variable "a" by adding the two equations together.
First, we need to make the coefficients of "a" in both equations equal in magnitude but opposite in sign. To do this, we can multiply the second equation by 7:
7(-7a + b) = 7(25)
-49a + 7b = 175
Now we have two equations:
7a + 6b = 3
-49a + 7b = 175
Adding the two equations together:
(7a + 6b) + (-49a + 7b) = 3 + 175
-42a + 13b = 178
Now, we have one equation with only the variable "b". We can solve for "b" by isolating it:
-42a + 13b = 178
13b = 42a + 178
b = (42a + 178) / 13
Now, substitute the expression for "b" into one of the original equations, for example, the first equation:
7a + 6((42a + 178) / 13) = 3
Now, solve for "a". Simplify the equation and isolate "a":
7a + (252a + 1068) / 13 = 3
Multiplying through by 13 to clear the fraction:
91a + 252a + 1068 = 39
343a = -1029
a = -3
Now, substitute the value of "a" back into one of the original equations, for example, the second equation, to find the value of "b":
-7(-3) + b = 25
21 + b = 25
b = 25 - 21
b = 4
Therefore, the solution to the system of equations is: a = -3 and b = 4.