Math

Question Determine the type of system of linear equations with equations y=12x52y=\frac{1}{2} x-\frac{5}{2} and y=x2y=x-2.

Studdy Solution

STEP 1

Assumptions
1. We have a system of two linear equations in two variables xx and yy.
2. Line 1 is given by y=12x52y=\frac{1}{2} x-\frac{5}{2}.
3. Line 2 is given by y=x2y=x-2.
4. We need to determine if the system is inconsistent, consistent dependent, or consistent independent.

STEP 2

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

STEP 3

Identify the slope and y-intercept of Line 1 from its equation.
The slope-intercept form of a line is y=mx+by=mx+b, where mm is the slope and bb is the y-intercept.

STEP 4

For Line 1, the slope m1m_1 is 12\frac{1}{2} and the y-intercept b1b_1 is 52-\frac{5}{2}.

STEP 5

Identify the slope and y-intercept of Line 2 from its equation.

STEP 6

For Line 2, the slope m2m_2 is 11 and the y-intercept b2b_2 is 2-2.

STEP 7

Compare the slopes m1m_1 and m2m_2 of Line 1 and Line 2.

STEP 8

Since m1=12m_1 = \frac{1}{2} and m2=1m_2 = 1, and 121\frac{1}{2} \neq 1, the slopes are not equal.

STEP 9

Because the slopes are not equal, the lines are not parallel and they will intersect at exactly one point.

STEP 10

Since the lines intersect at exactly one point, the system of equations is consistent and independent.
The system of equations is consistent independent.

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