Math  /  Algebra

QuestionLine 1: y=3x+4y=-3 x+4
Line 2: 3x+y=43 x+y=4
This system of equations is: consistent independent consistent dependent inconsistent This means the system has: a unique solution Solution: \square \square infinitely many solutions no solution
Line 1: y=3x3y=3 x-3
Line 2: y=3x+2y=3 x+2
This system of equations is: consistent independent consistent dependent inconsistent This means the system has: a unique solution Solution: \square \square infinitely many solutions no solution
Line 1: y=2x+1y=-2 x+1 Line 2: y=12x+1y=-\frac{1}{2} x+1
This system of equations is: consistent independent consistent dependent inconsistent This means the system has: a unique solution Solution: \square ( \square \square infinitely many solutions no solution

Studdy Solution

STEP 1

1. A system of linear equations can be classified as consistent independent, consistent dependent, or inconsistent.
2. Consistent independent systems have exactly one solution (unique solution).
3. Consistent dependent systems have infinitely many solutions.
4. Inconsistent systems have no solution.

STEP 2

1. Analyze the first system of equations.
2. Analyze the second system of equations.
3. Analyze the third system of equations.

STEP 3

Analyze the first system of equations:
Line 1: y=3x+4 y = -3x + 4
Line 2: 3x+y=4 3x + y = 4
Convert Line 2 to the slope-intercept form y=mx+b y = mx + b :
y=3x+4 y = -3x + 4
Notice that both lines have the same equation. Therefore, they are the same line.

STEP 4

Since both lines are the same, the system is consistent dependent.
This means the system has infinitely many solutions.
Solution: \infty

STEP 5

Analyze the second system of equations:
Line 1: y=3x3 y = 3x - 3
Line 2: y=3x+2 y = 3x + 2
Both lines have the same slope but different y-intercepts.

STEP 6

Since the lines are parallel (same slope, different intercepts), the system is inconsistent.
This means the system has no solution.
Solution: \emptyset

STEP 7

Analyze the third system of equations:
Line 1: y=2x+1 y = -2x + 1
Line 2: y=12x+1 y = -\frac{1}{2}x + 1
The lines have different slopes.

STEP 8

Since the lines have different slopes, they intersect at exactly one point.
The system is consistent independent.
This means the system has a unique solution.
Solution: (x,y)(x, y) where the lines intersect.

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