Question

How many solutions does this linear system have? 

one solution: (8, 0)one solution: (0, 8)no solution infinite number of solutions

Answer 1

The solution set of a linear system is determined by the relationships between the equations given. Analyzing the slope and intercept of the lines, we can identify if the system has one solution, no solution, or an infinite number of solutions.

Step-by-step explanation:

The number of solutions a linear system has can be determined by analyzing the equations involved. If the equations represent different lines, there is one point of intersection (one solution); if they represent the same line, there are infinitely many points of intersection (infinite number of solutions); and if they represent parallel, non-overlapping lines, there is no point of intersection (no solution).

Given the equations '7y = 6x + 8', '4y = 8', and 'y + 7 = 3x', we need to put them into slope-intercept form (y = mx + b) and compare their slopes (m) and y-intercepts (b). For example, from '4y = 8', we can immediately see that this represents the line y = 2, which is a horizontal line. By analyzing similarly, we can conclude about the number of solutions the system has based on whether these lines intersect at a single point, overlap completely, or never intersect.

Answer 2

Solve the second equation for y

x+2y = -8
x+2y-x = -8-x
2y = -x-8
2y/2 = -x/2 -8/2
y = (-1/2)x - 4

The slope here is -1/2, which is the same slope as the first equation. Because of this fact and the fact that the y intercept is not the same, we have two parallel lines. The parallel lines never cross.

There are no solutions