If -1,2 is the solution to the following linear system, what are the values of the coefficents a and b . Show your work to support your answer
2ax-3by= - 14
5ax - 2by = -2
HOW WOULD I SOLVE
substutx x and y into the equations:
-2a-6b=-14
-5a-20b=-2
then solve for a and b.
x = -1 and y = 2
just plug in the values for x and y, and you get
-2a - 6b = - 14
-5a - 4b = -2
Now multiply by 2 on the top and 3 on the bottom to get
-4a - 12b = -28
-15a - 12b = -6
Now subtract and y is eliminated:
11a = -22
a = -2
and now use that to find b.
To find the values of the coefficients a and b, we can substitute the given solution (-1,2) into the given system of equations and solve for a and b. Here's how you can do it step by step:
Step 1: Substitute the given values for x and y into the first equation:
2ax - 3by = -14
2a(-1) - 3b(2) = -14
-2a - 6b = -14
Step 2: Simplify the equation obtained in step 1:
-2a - 6b = -14
Step 3: Substitute the given values for x and y into the second equation:
5ax - 2by = -2
5a(-1) - 2b(2) = -2
-5a - 4b = -2
Step 4: Simplify the equation obtained in step 3:
-5a - 4b = -2
Step 5: We now have a system of two linear equations:
-2a - 6b = -14
-5a - 4b = -2
Step 6: Solve the system of equations using any appropriate method, such as substitution or elimination. For illustration, we will use the method of substitution:
From equation 2, solve for a:
-5a = -2 + 4b
a = (2 - 4b)/5
Substitute this value of a into equation 1:
-2(2 - 4b)/5 - 6b = -14
-4(2 - 4b) - 30b = -70
-8 + 16b - 30b = -70
-14b = -62
b = -62 / -14
b = 31/7
Step 7: Substitute the value of b back into equation 2 to find the value of a:
-5a - 4(31/7) = -2
-5a - 124/7 = -2
-5a = -2 + 124/7
-5a = -14/7 + 124/7
-5a = 110/7
a = (110/7) / -5
a = 110 / (7 * -5)
a = -22/7
Therefore, the values of the coefficients a and b are:
a = -22/7
b = 31/7
You can now verify these values by substituting them back into the original equations and checking if they satisfy the given system.