Question

Need to know the right way to write out the equations/where to graph which points.

Answer 1

Answer:

1. $y = -\frac{5}{4}x + \frac{35}{2}$

2. $y = \frac{15}{13}x - \frac{4}{13}$

3. $y = -\frac{1}{3}x + 175$

4. $y = \frac{1}{8}x + \frac{75}{2}$

Explanation:

The graphs are straight, so they are linear & can be written in "slope-intercept form" y = mx + b. Slope-intercept form is a linear equation using the slope "m" and y-intercept "b"

The slope is how steep the line is and if it goes up (positive slope) or down (negative slope)

The y-intercept is when the graph hits the y-axis

FORMULAS:

Slope-intercept form: y = mx + b

Slope = $m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

STEPS in each question:

*Choose point 1 and point 2. Points are written (x, y).

*Calculate slope "m"

*Calculate y-intercept "b"

*Put the equation in slope-intercept form

*Graph by plotting points

QUESTION 1

Write points 1 and 2

Point 1: (2, 15)    x₁ = 2   y₁ = 15

Point 2: (10, 5)   x₂ = 10  y₂ = 5

Calculate slope using the points

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

$m = \frac{5-15}{10-2}$   Subtract

$m = \frac{-10}{8}$    Reduce to lowest terms. -10 and 8 are divisible by 2

$m = \frac{-5}{4}$  Fix the formatting. The "-" is usually written outside.

$m = -\frac{5}{4}$  Slope Notice negative slope is downwards graph

Calculate y-intercept "b" using slope and any point

m = -5/4; I will use point 1. x = 2; y = 15

Substitute the information into slope-intercept form

y = mx + b

15 = (-5/4)(2) + b   Isolate "b" by separating it from other numbers

15 = -10/4 + b

15 = -5/2 + b

b = 15 + 5/2   Add 5/2 to both sides

b = 35/2   y-intercept

Write the equation with slope-intercept form, replacing "m" and "b".

$y = -\frac{5}{4}x + \frac{35}{2}$

QUESTION 2

Choose point 1 and 2

Point 1: (15, 17)  x₁ = 15   y₁ = 17

Point 2: (2, 2)   x₂ = 2    y₂ = 2

Calculate "m" with the points in the slope formula

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$    Substitute the points

$m = \frac{2-17}{2-15}$     Subtract

$m = \frac{-15}{-13}$    Dividing two negatives make a positive

$m = \frac{15}{13}$   Slope Positive slope for upwards graph

Calculate y-intercept "b"

Use m = 15/13; x₂ = 2; y₂ = 2

y = mx + b

2 = (15/13)(2) + b  Multiply

2 = 30/13 + b

b = 2 - 30/13  Subtract 30/13 from both sides

b = 26/13 - 30/13

b = -4/13   y-intercept

Write the equation

$y = \frac{15}{13}x - \frac{4}{13}$

QUESTION 3

Write points 1 and 2

The demand means what the people want to buy. In the question, only take sections that say "people that are willing and able to buy".

"sell for $75, there are 150,000 people buy"

"price falls to $50, 75,000 people more buy" 75,000 MORE means 75,000+150,000.

Point are written (x, y). "x" is quantity because it's on the x-axis. "y" is price because it's on the y-axis

Since the number of people is so large, you can write them in 000s (divide numbers by 1000)

Point 1: (150, 75)      x₁ = 150   y₁ = 75

Point 2: (225, 50)    x₂ = 225    y₂ = 50

Calculate slope

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$  Substitute the points

$m = \frac{50-75}{225-150}$   Subtract

$m = \frac{-25}{75}$  Top and bottom numbers are divisible by 25

$m = -\frac{1}{3}$  Slope

Calculate "b"

Use m = -1/3    x₂ = 225    y₂ = 50

y = mx + b   Substitute the slope and coordinates of a point

50 = (-1/3)(225) + b   Simplify the multiplication

50 = -75 + b   Isolate "b" by adding 75 from both sides

b = 125   y-intercept

Do the equation

$y = -\frac{1}{3}x + 175$

Graph by plotting points

The numbers of people are so big, so I made the scale for quantity in 1000s. Beside "Quantity" on the graph, write (in 000s). The Quantity scale can count by 50s

Plot these points

Point 1 (150, 75)

Point 2 (225, 50)

Connect the dots

QUESTION 4

Choose point 1 and point 2

The supply means the number of tickets that the producers are wiling to release.

"sell for $75 ... producers release 300,000 tickets"

"price falls to $50 ... producers release 100,000 tickets"

"x" is quantity and "y" is price because of the axis they are on

Since the number of people is so large, you can write them in 000s (divide numbers by 1000)

Point 1 (300, 75)   x₁ = 300  y₁ = 75

Point 2 (100, 50)   x₂ = 100   y₂ = 50

Calculate "m"

$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$    Substitute the points

$m = \frac{50-75}{100-300}$    Subtract

$m = \frac{-25}{-200}$   Top and bottom numbers divide by -25

$m = \frac{1}{8}$    Slope

Calculate "b"

Use m = 1/8; x₁ = 300; y₁ = 75

y = mx + b   Substitute "m"; x₁ and y₁

75 = (1/8)(300) + b   Simplify by multiplying

75 = 37.5 + b    Subtract 37.5 from both sides

b = 37.5 OR 75/2   y-intercept

Write the equation

Use m = 1/8 & b = 75/2

$y = \frac{1}{8}x + \frac{75}{2}$

Graph the points

The points are:

Point 1 (300, 75)

Point 2 (100, 50)

Plot them as (x, y)

The Quantity counts by 1000s since the number of tickets released is very large.