Math / Geometry

Question(4) Find the slope of each line described below. (a) The line through $(3,7)$ and $(-1,4)$ . (b) The line passing through the two points, $(-3,2)$ and $(6,2)$ . (c) A vertical line. (d) A horizontal line.

Studdy Solution

STEP 1

What is this asking? We need to find the steepness (slope) of different lines, given some info about them. Watch out! Remember, vertical lines have an undefined slope, not a slope of zero!

STEP 2

1. Slope through two points (a)
2. Slope through two points (b)
3. Slope of a vertical line (c)
4. Slope of a horizontal line (d)

STEP 3

Alright, let's find the slope of the line going through the points $(3,7)$ and $(-1,4)$ !
Remember, slope is just how much the line goes up or down, divided by how much it goes across.
It's like rise over run!

STEP 4

We can define our two points as $(x_1, y_1) = (3,7)$ and $(x_2, y_2) = (-1,4)$ .
Now, we can plug these values into our trusty slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

STEP 5

Let's substitute our values: $m = \frac{4 - 7}{-1 - 3}$

STEP 6

Calculate the differences: $m = \frac{-3}{-4}$

STEP 7

Simplify the fraction by dividing both the numerator and denominator by $-1$ : $m = \frac{3}{4}$ So, the slope of this line is $\frac{3}{4}$ !

STEP 8

Now, let's tackle the line through $(-3,2)$ and $(6,2)$ .

STEP 9

Define the points as $(x_1, y_1) = (-3,2)$ and $(x_2, y_2) = (6,2)$ .
Using the same slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

STEP 10

Substitute the values: $m = \frac{2 - 2}{6 - (-3)}$

STEP 11

Calculate the differences: $m = \frac{0}{9}$

STEP 12

Simplify: $m = 0$ A slope of zero!
That means it's a horizontal line!

STEP 13

Think about a vertical line.
It goes straight up and down, meaning it doesn't move horizontally at all.

STEP 14

Since slope is rise over run, and the "run" (change in $x$ ) is zero for a vertical line, we'd be dividing by zero.
And we can't do that!
So, the slope of a vertical line is undefined.

STEP 15

A horizontal line doesn't go up or down at all.
It only moves horizontally.

STEP 16

So, the "rise" (change in $y$ ) is zero.
Since slope is rise over run, we have zero divided by something that isn't zero.
And zero divided by anything (except zero) is just zero!
So, the slope of a horizontal line is always zero.

STEP 17

(a) The slope of the line through $(3,7)$ and $(-1,4)$ is $\frac{3}{4}$ . (b) The slope of the line through $(-3,2)$ and $(6,2)$ is $0$ . (c) The slope of a vertical line is undefined. (d) The slope of a horizontal line is $0$ .

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Question(4) Find the slope of each line described below. (a) The line through $(3,7)$ and $(-1,4)$ . (b) The line passing through the two points, $(-3,2)$ and $(6,2)$ . (c) A vertical line. (d) A horizontal line.

Studdy Solution

STEP 1

What is this asking? We need to find the steepness (slope) of different lines, given some info about them. Watch out! Remember, vertical lines have an undefined slope, not a slope of zero!

STEP 2

1. Slope through two points (a)
2. Slope through two points (b)
3. Slope of a vertical line (c)
4. Slope of a horizontal line (d)

STEP 3

Alright, let's find the slope of the line going through the points $(3,7)$ and $(-1,4)$ !
Remember, slope is just how much the line goes up or down, divided by how much it goes across.
It's like rise over run!

STEP 4

We can define our two points as $(x_1, y_1) = (3,7)$ and $(x_2, y_2) = (-1,4)$ .
Now, we can plug these values into our trusty slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

STEP 5

Let's substitute our values: $m = \frac{4 - 7}{-1 - 3}$

STEP 6

Calculate the differences: $m = \frac{-3}{-4}$

STEP 7

Simplify the fraction by dividing both the numerator and denominator by $-1$ : $m = \frac{3}{4}$ So, the slope of this line is $\frac{3}{4}$ !

STEP 8

Now, let's tackle the line through $(-3,2)$ and $(6,2)$ .

STEP 9

Define the points as $(x_1, y_1) = (-3,2)$ and $(x_2, y_2) = (6,2)$ .
Using the same slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

STEP 10

Substitute the values: $m = \frac{2 - 2}{6 - (-3)}$

STEP 11

Calculate the differences: $m = \frac{0}{9}$

STEP 12

Simplify: $m = 0$ A slope of zero!
That means it's a horizontal line!

STEP 13

Think about a vertical line.
It goes straight up and down, meaning it doesn't move horizontally at all.

STEP 14

Since slope is rise over run, and the "run" (change in $x$ ) is zero for a vertical line, we'd be dividing by zero.
And we can't do that!
So, the slope of a vertical line is undefined.

STEP 15

A horizontal line doesn't go up or down at all.
It only moves horizontally.

STEP 16

So, the "rise" (change in $y$ ) is zero.
Since slope is rise over run, we have zero divided by something that isn't zero.
And zero divided by anything (except zero) is just zero!
So, the slope of a horizontal line is always zero.

STEP 17

(a) The slope of the line through $(3,7)$ and $(-1,4)$ is $\frac{3}{4}$ . (b) The slope of the line through $(-3,2)$ and $(6,2)$ is $0$ . (c) The slope of a vertical line is undefined. (d) The slope of a horizontal line is $0$ .

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Question(4) Find the slope of each line described below. (a) The line through (3,7)(3,7)(3,7) and (−1,4)(-1,4)(−1,4). (b) The line passing through the two points, (−3,2)(-3,2)(−3,2) and (6,2)(6,2)(6,2). (c) A vertical line. (d) A horizontal line.

Studdy Solution

STEP 1

What is this asking? We need to find the steepness (slope) of different lines, given some info about them. Watch out! Remember, vertical lines have an undefined slope, not a slope of zero!

STEP 2

1. Slope through two points (a)2. Slope through two points (b)3. Slope of a vertical line (c)4. Slope of a horizontal line (d)

STEP 3

Alright, let's find the slope of the line going through the points (3,7)(3,7)(3,7) and (−1,4)(-1,4)(−1,4)!Remember, slope is just how much the line goes up or down, divided by how much it goes across.It's like rise over run!

STEP 4

We can **define** our two points as (x1,y1)=(3,7) (x_1, y_1) = (3,7) (x1​,y1​)=(3,7) and (x2,y2)=(−1,4) (x_2, y_2) = (-1,4) (x2​,y2​)=(−1,4).Now, we can plug these values into our trusty slope formula: m=y2−y1x2−x1 m = \frac{y_2 - y_1}{x_2 - x_1} m=x2​−x1​y2​−y1​​

STEP 5

Let's **substitute** our values: m=4−7−1−3 m = \frac{4 - 7}{-1 - 3} m=−1−34−7​

STEP 6

**Calculate** the differences: m=−3−4 m = \frac{-3}{-4} m=−4−3​

STEP 7

**Simplify** the fraction by dividing both the numerator and denominator by −1-1−1: m=34 m = \frac{3}{4} m=43​ So, the slope of this line is 34\frac{3}{4}43​!

STEP 8

Now, let's tackle the line through (−3,2)(-3,2)(−3,2) and (6,2)(6,2)(6,2).

STEP 9

**Define** the points as (x1,y1)=(−3,2) (x_1, y_1) = (-3,2) (x1​,y1​)=(−3,2) and (x2,y2)=(6,2) (x_2, y_2) = (6,2) (x2​,y2​)=(6,2).Using the same slope formula: m=y2−y1x2−x1 m = \frac{y_2 - y_1}{x_2 - x_1} m=x2​−x1​y2​−y1​​

STEP 10

**Substitute** the values: m=2−26−(−3) m = \frac{2 - 2}{6 - (-3)} m=6−(−3)2−2​

STEP 11

**Calculate** the differences: m=09 m = \frac{0}{9} m=90​

STEP 12

**Simplify**: m=0 m = 0 m=0 A slope of zero!That means it's a horizontal line!

STEP 13

Think about a vertical line.It goes straight up and down, meaning it doesn't move horizontally at all.

STEP 14

Since slope is rise over run, and the "run" (change in xxx) is zero for a vertical line, we'd be dividing by zero.And we can't do that!So, the slope of a vertical line is **undefined**.

STEP 15

A horizontal line doesn't go up or down at all.It only moves horizontally.

STEP 16

So, the "rise" (change in yyy) is zero.Since slope is rise over run, we have zero divided by something that isn't zero.And zero divided by anything (except zero) is just zero!So, the slope of a horizontal line is always **zero**.

STEP 17

(a) The slope of the line through (3,7)(3,7)(3,7) and (−1,4)(-1,4)(−1,4) is 34\frac{3}{4}43​. (b) The slope of the line through (−3,2)(-3,2)(−3,2) and (6,2)(6,2)(6,2) is 000. (c) The slope of a vertical line is undefined. (d) The slope of a horizontal line is 000.

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Question(4) Find the slope of each line described below. (a) The line through $(3,7)$ and $(-1,4)$ . (b) The line passing through the two points, $(-3,2)$ and $(6,2)$ . (c) A vertical line. (d) A horizontal line.

1. Slope through two points (a)
2. Slope through two points (b)
3. Slope of a vertical line (c)
4. Slope of a horizontal line (d)

Alright, let's find the slope of the line going through the points $(3,7)$ and $(-1,4)$ !
Remember, slope is just how much the line goes up or down, divided by how much it goes across.
It's like rise over run!

We can define our two points as $(x_1, y_1) = (3,7)$ and $(x_2, y_2) = (-1,4)$ .
Now, we can plug these values into our trusty slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Let's substitute our values: $m = \frac{4 - 7}{-1 - 3}$

Calculate the differences: $m = \frac{-3}{-4}$

Simplify the fraction by dividing both the numerator and denominator by $-1$ : $m = \frac{3}{4}$ So, the slope of this line is $\frac{3}{4}$ !

Now, let's tackle the line through $(-3,2)$ and $(6,2)$ .

Define the points as $(x_1, y_1) = (-3,2)$ and $(x_2, y_2) = (6,2)$ .
Using the same slope formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$

Substitute the values: $m = \frac{2 - 2}{6 - (-3)}$

Calculate the differences: $m = \frac{0}{9}$

Simplify: $m = 0$ A slope of zero!
That means it's a horizontal line!

Think about a vertical line.
It goes straight up and down, meaning it doesn't move horizontally at all.

Since slope is rise over run, and the "run" (change in $x$ ) is zero for a vertical line, we'd be dividing by zero.
And we can't do that!
So, the slope of a vertical line is undefined.

A horizontal line doesn't go up or down at all.
It only moves horizontally.

So, the "rise" (change in $y$ ) is zero.
Since slope is rise over run, we have zero divided by something that isn't zero.
And zero divided by anything (except zero) is just zero!
So, the slope of a horizontal line is always zero.

(a) The slope of the line through $(3,7)$ and $(-1,4)$ is $\frac{3}{4}$ . (b) The slope of the line through $(-3,2)$ and $(6,2)$ is $0$ . (c) The slope of a vertical line is undefined. (d) The slope of a horizontal line is $0$ .