Math / Algebra

Question1. ¿Qué recta tiene una pendiente de $\frac{3}{2}$ ? (A) (B) (C) (D)
2. Halla la pendiente de la recta.
3. Selecciona todos los enunciados verdaderos sobre una relación lineal que también es una relación proporcional. $\square$ La gráfica pasa por el origen. La gráfica no pasa por el origen. La constante de proporcionalidad se puede hallar usando $\frac{\text { distancia horizontal }}{\text { distancia vertical }}$ La pendiente es igual que la constante de proporcionalidad. $\qquad$ La pendiente es igual que la tasa por unidad.
4. La gráfica muestra la distancia recorrida por un velocista de nivel olímpico, $y$ , a lo largo del tiempo durante una carrera, $x$ . Halla la pendiente de la recta y completa las siguientes oraciones. $\square$ $\square$ el tiempo en que corre el velocista La pendiente representa $\square$ la distancia que recorre el velocista la velocidad del velocista en metros por segundo
5. Los puntos $(80,20)$ y $(120,30)$ forman una relación proporcional. ¿Cuál es la pendiente de la recta que pasa por estos puntos? (A) $\frac{1}{4}$ (B) 4 (C) 10
Carrera de 100 metros

Studdy Solution

STEP 1

What is this asking? We need to find a line with a specific steepness, find the steepness of another line, identify true statements about proportional relationships, find the steepness of a line on a graph and interpret it, and calculate the steepness of a line passing through two given points. Watch out! Don't mix up horizontal and vertical distances when calculating steepness!
Also, remember that proportional relationships always go through the origin!

STEP 2

1. Find the Line
2. Calculate Steepness
3. Proportional Relationships
4. Sprinter's Speed
5. Proportional Points

STEP 3

We're looking for a line with a steepness, or slope, of $\frac{3}{2}$ .
Remember, slope is "rise over run," or how much the line goes up divided by how much it goes across.

STEP 4

A slope of $\frac{3}{2}$ means that for every 2 units we move to the right, the line goes up 3 units.
Look for this line among the choices!

STEP 5

To find the slope, we need two points on the line.
Pick two points that are easy to read from the graph.

STEP 6

Let's say we pick the points $(x_1, y_1)$ and $(x_2, y_2)$ .
The slope is calculated as $\frac{y_2 - y_1}{x_2 - x_1}$ , which is the change in the vertical direction divided by the change in the horizontal direction.
Plug in your points and calculate!

STEP 7

A proportional relationship means that as one thing changes, the other changes by a constant factor.
This constant factor is also the slope!

STEP 8

Because proportional relationships always pass through the origin $(0,0)$ , the first statement is true, and the second is false.

STEP 9

The constant of proportionality is found using $\frac{\text{vertical distance}}{\text{horizontal distance}}$ , not the other way around, so the third statement is false.

STEP 10

The slope is equal to the constant of proportionality, and the slope is the rate per unit (how much y changes for every one unit change in x).
So, the last two statements are true!

STEP 11

Pick two points on the graph.
Let's say we choose $(x_1, y_1)$ and $(x_2, y_2)$ .

STEP 12

Calculate the slope as $\frac{y_2 - y_1}{x_2 - y_1}$ .
This slope represents the change in distance (vertical axis) divided by the change in time (horizontal axis).

STEP 13

Since distance divided by time is speed, the slope represents the sprinter's speed in meters per second!

STEP 14

We have two points: $(80, 20)$ and $(120, 30)$ .
Let $(80, 20)$ be $(x_1, y_1)$ and $(120, 30)$ be $(x_2, y_2)$ .

STEP 15

The slope is $\frac{y_2 - y_1}{x_2 - x_1} = \frac{30 - 20}{120 - 80} = \frac{10}{40} = \frac{1}{4}$ .

STEP 16

1. The line with a slope of $\frac{3}{2}$ depends on the provided graphs.
2. The slope of the line depends on the provided graph.
3. True statements: The graph passes through the origin, the slope is equal to the constant of proportionality, and the slope is equal to the rate per unit.
4. The slope represents the sprinter's speed in meters per second.
The value of the slope depends on the provided graph.
5. The slope of the line passing through the points $(80, 20)$ and $(120, 30)$ is $\frac{1}{4}$ .

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Studdy Solution

STEP 1

STEP 2

1. Find the Line
2. Calculate Steepness
3. Proportional Relationships
4. Sprinter's Speed
5. Proportional Points

STEP 3

We're looking for a line with a steepness, or slope, of $\frac{3}{2}$ .
Remember, slope is "rise over run," or how much the line goes up divided by how much it goes across.

STEP 4

A slope of $\frac{3}{2}$ means that for every 2 units we move to the right, the line goes up 3 units.
Look for this line among the choices!

STEP 5

To find the slope, we need two points on the line.
Pick two points that are easy to read from the graph.

STEP 6

Let's say we pick the points $(x_1, y_1)$ and $(x_2, y_2)$ .
The slope is calculated as $\frac{y_2 - y_1}{x_2 - x_1}$ , which is the change in the vertical direction divided by the change in the horizontal direction.
Plug in your points and calculate!

STEP 7

A proportional relationship means that as one thing changes, the other changes by a constant factor.
This constant factor is also the slope!

STEP 8

Because proportional relationships always pass through the origin $(0,0)$ , the first statement is true, and the second is false.

STEP 9

The constant of proportionality is found using $\frac{\text{vertical distance}}{\text{horizontal distance}}$ , not the other way around, so the third statement is false.

STEP 10

The slope is equal to the constant of proportionality, and the slope is the rate per unit (how much y changes for every one unit change in x).
So, the last two statements are true!

STEP 11

Pick two points on the graph.
Let's say we choose $(x_1, y_1)$ and $(x_2, y_2)$ .

STEP 12

Calculate the slope as $\frac{y_2 - y_1}{x_2 - y_1}$ .
This slope represents the change in distance (vertical axis) divided by the change in time (horizontal axis).

STEP 13

Since distance divided by time is speed, the slope represents the sprinter's speed in meters per second!

STEP 14

We have two points: $(80, 20)$ and $(120, 30)$ .
Let $(80, 20)$ be $(x_1, y_1)$ and $(120, 30)$ be $(x_2, y_2)$ .

STEP 15

The slope is $\frac{y_2 - y_1}{x_2 - x_1} = \frac{30 - 20}{120 - 80} = \frac{10}{40} = \frac{1}{4}$ .

STEP 16

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Studdy Solution

STEP 1

STEP 2

1. Find the Line2. Calculate Steepness3. Proportional Relationships4. Sprinter's Speed5. Proportional Points

STEP 3

We're looking for a line with a steepness, or *slope*, of 32\frac{3}{2}23​.Remember, slope is "rise over run," or how much the line goes up divided by how much it goes across.

STEP 4

A slope of 32\frac{3}{2}23​ means that for every **2** units we move to the right, the line goes up **3** units.Look for this line among the choices!

STEP 5

To find the slope, we need two points on the line.Pick two points that are easy to read from the graph.

STEP 6

STEP 7

A **proportional relationship** means that as one thing changes, the other changes by a constant factor.This constant factor is also the slope!

STEP 8

Because proportional relationships always pass through the origin (0,0)(0,0)(0,0), the first statement is **true**, and the second is **false**.

STEP 9

The constant of proportionality is found using vertical distancehorizontal distance\frac{\text{vertical distance}}{\text{horizontal distance}}horizontal distancevertical distance​, not the other way around, so the third statement is **false**.

STEP 10

The slope *is* equal to the constant of proportionality, and the slope *is* the rate per unit (how much *y* changes for every one unit change in *x*).So, the last two statements are **true**!

STEP 11

Pick two points on the graph.Let's say we choose (x1,y1)(x_1, y_1)(x1​,y1​) and (x2,y2)(x_2, y_2)(x2​,y2​).

STEP 12

Calculate the slope as y2−y1x2−y1\frac{y_2 - y_1}{x_2 - y_1}x2​−y1​y2​−y1​​.This slope represents the change in distance (vertical axis) divided by the change in time (horizontal axis).

STEP 13

Since distance divided by time is **speed**, the slope represents the sprinter's speed in meters per second!

STEP 14

We have two points: (80,20)(80, 20)(80,20) and (120,30)(120, 30)(120,30).Let (80,20)(80, 20)(80,20) be (x1,y1)(x_1, y_1)(x1​,y1​) and (120,30)(120, 30)(120,30) be (x2,y2)(x_2, y_2)(x2​,y2​).

STEP 15

The slope is y2−y1x2−x1=30−20120−80=1040=14\frac{y_2 - y_1}{x_2 - x_1} = \frac{30 - 20}{120 - 80} = \frac{10}{40} = \frac{1}{4}x2​−x1​y2​−y1​​=120−8030−20​=4010​=41​.

STEP 16

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1. Find the Line
2. Calculate Steepness
3. Proportional Relationships
4. Sprinter's Speed
5. Proportional Points

We're looking for a line with a steepness, or slope, of $\frac{3}{2}$ .
Remember, slope is "rise over run," or how much the line goes up divided by how much it goes across.

A slope of $\frac{3}{2}$ means that for every 2 units we move to the right, the line goes up 3 units.
Look for this line among the choices!

To find the slope, we need two points on the line.
Pick two points that are easy to read from the graph.

A proportional relationship means that as one thing changes, the other changes by a constant factor.
This constant factor is also the slope!

Because proportional relationships always pass through the origin $(0,0)$ , the first statement is true, and the second is false.

The constant of proportionality is found using $\frac{\text{vertical distance}}{\text{horizontal distance}}$ , not the other way around, so the third statement is false.

The slope is equal to the constant of proportionality, and the slope is the rate per unit (how much y changes for every one unit change in x).
So, the last two statements are true!

Pick two points on the graph.
Let's say we choose $(x_1, y_1)$ and $(x_2, y_2)$ .

Calculate the slope as $\frac{y_2 - y_1}{x_2 - y_1}$ .
This slope represents the change in distance (vertical axis) divided by the change in time (horizontal axis).

Since distance divided by time is speed, the slope represents the sprinter's speed in meters per second!

We have two points: $(80, 20)$ and $(120, 30)$ .
Let $(80, 20)$ be $(x_1, y_1)$ and $(120, 30)$ be $(x_2, y_2)$ .

The slope is $\frac{y_2 - y_1}{x_2 - x_1} = \frac{30 - 20}{120 - 80} = \frac{10}{40} = \frac{1}{4}$ .