Final answer:
To solve system [a], you can use the method of substitution or elimination. The solution to system [a] is x = -15 and y = 3. To solve system [b], you can use the same method. The two equations in system [b] have infinitely many solutions and are dependent. The solution is any point that satisfies the equation, such as (x, y) = (3, 0).
Step-by-step explanation:
System [a]:
4x + 16y = 12
x + 2y = -9
System [b]:
4x + 16y = 12
x + 4y = 3
To solve system [a], we can use the method of substitution or elimination. Let's use elimination:
Multiply the second equation by 2 to match the coefficients of x. This gives:
2x + 4y = -18
Now subtract the second equation from the first equation:
(4x + 16y) - (2x + 4y) = 12 - (-18)
2x + 12y = 30
Simplify the equation:
x + 6y = 15
This new equation can be used with the original second equation to solve for x and y. Substitute the value of (x + 2y) from the second equation into the new equation:
(x + 2y) + 6y = 15
Simplify the equation:
x + 8y = 15
Now we have a system of two equations:
x + 8y = 15
x + 4y = 3
We can subtract the second equation from the first equation to eliminate x:
(x + 8y) - (x + 4y) = 15 - 3
4y = 12
Divide both sides by 4:
y = 3
Now substitute the value of y back into one of the original equations to solve for x:
x + 2(3) = -9
x + 6 = -9
Subtract 6 from both sides:
x = -15
So the solution to system [a] is x = -15 and y = 3.
To solve system [b], we can use the same method. Let's use elimination:
Multiply the second equation by 2 to match the coefficients of x. This gives:
2x + 8y = 6
Now subtract the second equation from the first equation:
(4x + 16y) - (2x + 8y) = 12 - 6
2x + 8y = 6
Simplify the equation:
x + 4y = 3
This new equation can be used with the original second equation to solve for x and y. Substitute the value of (x + 4y) from the second equation into the new equation:
(x + 4y) + 4y = 3
Simplify the equation:
x + 8y = 3
Now we have a system of two equations:
x + 8y = 3
x + 4y = 3
The two equations are the same, so they have infinitely many solutions and are dependent. The solution is any point that satisfies the equation, such as (x, y) = (3, 0).