What is the solution of the system? Use elimination.
5x + 6y = 17
−10x − 9y = −43
multiply the first equation by 2
+10 x + 12y = 34
−10 x − 9 y = −43
--------------------------------- add
3 y = - 9
y = - 9/3 and so forth
غذذا
To solve the system of equations using the elimination method, we need to eliminate one variable by multiplying one or both of the equations by a constant. Let's start by eliminating the x variable.
We can do this by multiplying the first equation by 2 and the second equation by 1:
(2) * (5x + 6y) = (2) * (17)
2(5x) + 2(6y) = 34
10x + 12y = 34
(1) * (-10x - 9y) = (1) * (-43)
-10x - 9y = -43
Now, we can add the two equations together to eliminate the x variable:
(10x + 12y) + (-10x - 9y) = 34 + (-43)
10x - 10x + 12y - 9y = -9
3y = -9
Simplifying the equation further, we have:
3y = -9
Now we can solve for y by dividing both sides by 3:
(3y)/3 = (-9)/3
y = -3
Now that we have the value of y, we can substitute it back into one of the original equations to solve for x. Let's use the first equation:
5x + 6(-3) = 17
5x - 18 = 17
5x = 17 + 18
5x = 35
Dividing both sides by 5 to solve for x:
(5x)/5 = 35/5
x = 7
So, the solution to the system of equations is x = 7 and y = -3.
To solve the given system of equations using elimination, we can eliminate one of the variables so that we end up with a single equation in one variable. Here's how we can proceed:
Step 1: Multiply the second equation by 5 to make the coefficients of x in both equations equal.
- 5(−10x − 9y) = 5(−43)
Simplifying this equation gives: -50x - 45y = -215.
Step 2: Add the two equations together to eliminate x.
(5x + 6y) + (-50x - 45y) = 17 + (-215)
Simplifying this equation gives: -45y - 39y = -198
Step 3: Combine like terms and solve for y.
-85y = -198
Dividing both sides of the equation by -85, we find: y = 198/85.
Step 4: Substitute the value of y back into one of the original equations to solve for x.
5x + 6(198/85) = 17
Multiplying 6 and 198/85 gives: 1188/85.
Subtracting 1188/85 from both sides gives: 5x = 17 - 1188/85
Subtracting the fractions and simplifying gives: 5x = (1445 - 1188)/85
Thus, 5x = 257/85.
Step 5: Solve for x by dividing both sides by 5.
x = (257/85) / 5
Divide: x = 257/85 * 1/5
Multiply the numerators: x = 257/425.
Therefore, the solution to the system of equations 5x + 6y = 17 and −10x − 9y = −43, using elimination, is x = 257/425 and y = 198/85.