Question

Part 1: Provide a system of TWO equations in slope-intercept form, with only one solution. Using complete sentences, explain why this system has one solution.

Part 2: Provide a system of TWO equations in slope-intercept form with no solutions. Using complete sentences, explain why this system has no solutions.

Part 3: Provide a system of TWO equations in slope-intercept form with infinitely many solutions. Using complete sentences, explain why this system has infinitely many solutions.

Answer 1

PART 1
y = x +7
y = 2x +3
solve x+7 = 2x+3
4 = x
y = 4+7 =11
check
y = 2*4 +3 =11 checks
answer (4,11) is the only point where the two lines intersect. two distinct non-parallel lines cross at exactly one point. This is called an "independent" system of equations, and the solution is always some x,y-point.

PART 2
y = x+7
y = x+3
solve x+7 = x+3
7= 3 no solution
the two lines never intersect. two distinct lines that are parallel. Since parallel lines never cross, then there can be no intersection; that is, for a system of equations that graphs as parallel lines, there can be no solution. This is called an "inconsistent" system of equations, and it has no solution.

PART 3
x=y
2x = 2y
any point on the line is a solution there is an infinite number of solutions. only one line. Actually, it's the same line drawn twice. These "two" lines, really being the same line, "intersect" at every point along their length. This is called a "dependent" system, and the "solution" is the whole line.