Solved on Jan 22, 2024

Determine if the given system of linear equations with $y=\frac{1}{2}x-4$ and $y=\frac{1}{2}x+3$ has a unique solution, no solution, or infinitely many solutions.

STEP 1

Assumptions
1. Line 1 is given by the equation $y = \frac{1}{2} x - 4$ .
2. Line 2 is given by the equation $y = \frac{1}{2} x + 3$ .
3. A system of equations is inconsistent if there are no solutions.
4. A system of equations is consistent independent if there is exactly one solution.
5. A system of equations is consistent dependent if there are infinitely many solutions.

STEP 2

To determine the type of system, we compare the slopes and y-intercepts of the two lines.

STEP 3

Identify the slope and y-intercept of Line 1 from its equation.
$Slope\, of\, Line\, 1 = \frac{1}{2}$ $Y-intercept\, of\, Line\, 1 = -4$

STEP 4

Identify the slope and y-intercept of Line 2 from its equation.
$Slope\, of\, Line\, 2 = \frac{1}{2}$ $Y-intercept\, of\, Line\, 2 = 3$

STEP 5

Compare the slopes of Line 1 and Line 2.
$Slope\, of\, Line\, 1 = Slope\, of\, Line\, 2$

STEP 6

Since the slopes are equal, determine if the y-intercepts are also equal.
$Y-intercept\, of\, Line\, 1 \neq Y-intercept\, of\, Line\, 2$

STEP 7

Because the slopes are equal and the y-intercepts are different, the lines are parallel and do not intersect.

STEP 8

Conclude that the system of equations is inconsistent because parallel lines do not meet and therefore have no solution.
This system of equations is inconsistent, which means the system has no solution.

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Determine if the given system of linear equations with $y=\frac{1}{2}x-4$ and $y=\frac{1}{2}x+3$ has a unique solution, no solution, or infinitely many solutions.

STEP 1

STEP 2

To determine the type of system, we compare the slopes and y-intercepts of the two lines.

STEP 3

Identify the slope and y-intercept of Line 1 from its equation.
$Slope\, of\, Line\, 1 = \frac{1}{2}$ $Y-intercept\, of\, Line\, 1 = -4$

STEP 4

Identify the slope and y-intercept of Line 2 from its equation.
$Slope\, of\, Line\, 2 = \frac{1}{2}$ $Y-intercept\, of\, Line\, 2 = 3$

STEP 5

Compare the slopes of Line 1 and Line 2.
$Slope\, of\, Line\, 1 = Slope\, of\, Line\, 2$

STEP 6

Since the slopes are equal, determine if the y-intercepts are also equal.
$Y-intercept\, of\, Line\, 1 \neq Y-intercept\, of\, Line\, 2$

STEP 7

Because the slopes are equal and the y-intercepts are different, the lines are parallel and do not intersect.

STEP 8

Conclude that the system of equations is inconsistent because parallel lines do not meet and therefore have no solution.
This system of equations is inconsistent, which means the system has no solution.

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Determine if the given system of linear equations with y=12x−4y=\frac{1}{2}x-4y=21​x−4 and y=12x+3y=\frac{1}{2}x+3y=21​x+3 has a unique solution, no solution, or infinitely many solutions.

STEP 1

STEP 2

To determine the type of system, we compare the slopes and y-intercepts of the two lines.

STEP 3

Identify the slope and y-intercept of Line 1 from its equation.Slope of Line 1=12Slope\, of\, Line\, 1 = \frac{1}{2}SlopeofLine1=21​ Y−intercept of Line 1=−4Y-intercept\, of\, Line\, 1 = -4Y−interceptofLine1=−4

STEP 4

Identify the slope and y-intercept of Line 2 from its equation.Slope of Line 2=12Slope\, of\, Line\, 2 = \frac{1}{2}SlopeofLine2=21​ Y−intercept of Line 2=3Y-intercept\, of\, Line\, 2 = 3Y−interceptofLine2=3

STEP 5

Compare the slopes of Line 1 and Line 2.Slope of Line 1=Slope of Line 2Slope\, of\, Line\, 1 = Slope\, of\, Line\, 2SlopeofLine1=SlopeofLine2

STEP 6

Since the slopes are equal, determine if the y-intercepts are also equal.Y−intercept of Line 1≠Y−intercept of Line 2Y-intercept\, of\, Line\, 1 \neq Y-intercept\, of\, Line\, 2Y−interceptofLine1=Y−interceptofLine2

STEP 7

Because the slopes are equal and the y-intercepts are different, the lines are parallel and do not intersect.

STEP 8

Conclude that the system of equations is inconsistent because parallel lines do not meet and therefore have no solution.This system of equations is inconsistent, which means the system has no solution.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Determine if the given system of linear equations with $y=\frac{1}{2}x-4$ and $y=\frac{1}{2}x+3$ has a unique solution, no solution, or infinitely many solutions.

Identify the slope and y-intercept of Line 1 from its equation.
$Slope\, of\, Line\, 1 = \frac{1}{2}$ $Y-intercept\, of\, Line\, 1 = -4$

Identify the slope and y-intercept of Line 2 from its equation.
$Slope\, of\, Line\, 2 = \frac{1}{2}$ $Y-intercept\, of\, Line\, 2 = 3$

Compare the slopes of Line 1 and Line 2.
$Slope\, of\, Line\, 1 = Slope\, of\, Line\, 2$

Since the slopes are equal, determine if the y-intercepts are also equal.
$Y-intercept\, of\, Line\, 1 \neq Y-intercept\, of\, Line\, 2$

Conclude that the system of equations is inconsistent because parallel lines do not meet and therefore have no solution.
This system of equations is inconsistent, which means the system has no solution.