Math

QuestionDetermine if the following lines are parallel, perpendicular, or neither: Line 1: 4x10y=84x - 10y = 8 Line 2: y=52x4y = -\frac{5}{2}x - 4 Line 3: 2y=5x+32y = -5x + 3

Studdy Solution

STEP 1

Assumptions1. The equations of the lines are given in the standard form ax+by=cax + by = c. . Two lines are parallel if their slopes are equal.
3. Two lines are perpendicular if the product of their slopes is -1.
4. The slope of a line given in the standard form ax+by=cax + by = c is a/b-a/b.

STEP 2

First, we need to find the slopes of the given lines. The slope of a line given in the form ax+by=cax + by = c is a/b-a/b.
For Line1, a=4a =4 and b=10b = -10, so the slope is a/b-a/b.
m1=ab=410m1 = -\frac{a}{b} = -\frac{4}{-10}

STEP 3

Calculate the slope of Line1.
m1=10=0.m1 = -\frac{}{-10} =0.

STEP 4

For Line2, the equation is already in the slope-intercept form y=mx+cy = mx + c, where mm is the slope. So, the slope of Line2 is 2-\frac{}{2}.
m2=2m2 = -\frac{}{2}

STEP 5

For Line3, we can rearrange the equation to the form y=mx+cy = mx + c to find the slope.
2y=5x+3    y=52x+322y = -5x +3 \implies y = -\frac{5}{2}x + \frac{3}{2}So, the slope of Line3 is 52-\frac{5}{2}.
m3=52m3 = -\frac{5}{2}

STEP 6

Now that we have the slopes of all three lines, we can compare them to determine if the lines are parallel, perpendicular, or neither.
First, compare the slopes of Line1 and Line2.
If m1=m2m1 = m2, the lines are parallel. If m1m2=1m1 \cdot m2 = -1, the lines are perpendicular.
m1=m2andm1m2=1m1 = m2 \quad \text{and} \quad m1 \cdot m2 = -1

STEP 7

Plug in the values for m1m1 and m2m2 and check the conditions.
0.4=52and0.452=10.4 = -\frac{5}{2} \quad \text{and} \quad0.4 \cdot -\frac{5}{2} = -1

STEP 8

The first equation is not true, so Line1 and Line2 are not parallel. The second equation is also not true, so Line1 and Line2 are not perpendicular. Therefore, Line1 and Line2 are neither parallel nor perpendicular.

STEP 9

Next, compare the slopes of Line and Line3.
If m=m3m = m3, the lines are parallel. If mm3=m \cdot m3 = -, the lines are perpendicular.
m=m3andmm3=m = m3 \quad \text{and} \quad m \cdot m3 = -

STEP 10

Plug in the values for mm and m3m3 and check the conditions.
0.4=52and0.452=0.4 = -\frac{5}{2} \quad \text{and} \quad0.4 \cdot -\frac{5}{2} = -

STEP 11

The first equation is not true, so Line and Line3 are not parallel. The second equation is also not true, so Line and Line3 are not perpendicular. Therefore, Line and Line3 are neither parallel nor perpendicular.

STEP 12

Finally, compare the slopes of Line2 and Line.
If m2=mm2 = m, the lines are parallel. If m2m=m2 \cdot m = -, the lines are perpendicular.
m2=mandm2m=m2 = m \quad \text{and} \quad m2 \cdot m = -

STEP 13

Plug in the values for m2m2 and m3m3 and check the conditions.
52=52and5252=-\frac{5}{2} = -\frac{5}{2} \quad \text{and} \quad -\frac{5}{2} \cdot -\frac{5}{2} = -

STEP 14

The first equation is true, so Line2 and Line3 are parallel. The second equation is not true, so Line2 and Line3 are not perpendicular.
In conclusion, Line is neither parallel nor perpendicular to Line2 or Line3, and Line2 is parallel to Line3.

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