Answer:
Explanation:
For a system of equations to have infinite solutions, the two equations must represent the same line or be proportional to each other.
Let's consider the first equation:
1. 5x - y = 8
Now, we'll rewrite the second equation in terms of a and b:
2. 20x + ay = b
To determine the values of a and b that make these equations equivalent, we can rewrite equation 2 by isolating the variable y:
2.1. ay = -20x + b
2.2. y = (-20/a)x + (b/a)
Now, we can see that for the system to have infinite solutions, the slopes of both equations must be equal. This means that the coefficient of x in equation 1 (which is 5) must be equal to the coefficient of x in equation 2.1 (which is -20/a). Therefore:
5 = -20/a
To solve for a:
a = -20/5
a = -4
Now that we've determined the value of a, let's focus on making the y-intercepts match as well. In equation 1, the y-intercept is 8 (i.e., when x = 0, y = 8). In equation 2.2, the y-intercept is b/a.
For the system to have infinite solutions, these y-intercepts must be equal:
8 = b/a
To solve for b:
b = 8a
Now, since we already found that a = -4, we can determine the value of b:
b = 8 * (-4)
b = -32
So, for the system of equations to have infinite solutions, a must be equal to -4, and b must be equal to -32.