Linear Equations in Two Variables

​2C: write linear equations in two variables given a table of values, a graph, and a verbal description

​A linear equation is of the form
y = mx + b
where m represents the slope and
b represents the y-intercept.

​The graph of a linear function is a straight line.

​Strategies for determining SLOPE:

​WORDS

each for every rate
per - cost per item/amount per time
constant rate

​Strategies for determining Y-INTERCEPT:

​

TABLE

* y-value when x = 0. IF x = 0 is not given in the table, calculate b:

1.  Determine the slope.

2.  Choose any ordered pair (x, y) from the table.

3.  Plug in the slope, x-value, and y-value into y = mx + b.

4.  Solve for b: multiply m and x, then use the inverse (opposite)
operation to move the product of m and x with y.

OR
Enter the table into Desmos, then on Line2 type in y₁ ~ mx₁ + b.

​x	y​
​-3	-130​
2​	0​
5​	78​
11​	234​

​You can choose any point (x, y) from the table.
Pick a pair of numbers that seem simple.
I am going to choose (2, 0). Plug into y = mx + b.
Replace y with 0, m with 26, and x with 2.

0 = 26(2) + b Multiply 26(2). Then solve for b.
0 = 52 + b Subtract 52 to get b by itself.
-52 = b

NOW, plug in m and b into y = mx + b.

y = 26x - 52

​Strategies for determining Y-INTERCEPT:

​GRAPH

Point where the line crosses the y-axis. Written (0, y-value).

​Strategies for determining Y-INTERCEPT:

WORDS

initial amount starts fee fixed cost
flat fee starting value

​The cost of dance lessons offered at Wayside are determined by the following:

• A $30 registration fee plus $45 per month

$30 registration FEE indicates the y-intercept,
so b = 30.

​Printer ink costs $18.99 per cartridge. The company that sells the ink charges a flat rate shipping fee of $7.95, no matter the number of cartridges purchased in a single order.

$7.95 flat rate shipping fee indicates the
y-intercept, so b = 30.

Multiple Choice

The table represents some points on the graph of a linear function.

Which function represents this relationship?

• $$y=-\frac{3}{2}x+4$$

• $$y=-\frac{3}{2}x-14$$

• $$y=-\frac{2}{3}x-9$$

• $$y=-\frac{2}{3}x-1$$

Dropdown

The graph of a linear function is shown on the grid.

The function that is best represented by this graph is:



g(x) = ​

Select answer

x + ​​

Select answer

Dropdown

Printer ink costs $26.95 per cartridge. The company that sells the ink charges a flat rate shipping fee of $5.25, no matter the number of cartridges purchased in a single order.



The linear equation that models the total cost, t, for a single order of c cartridges is

​

Select answer

= ​

Select answer

​

Select answer

+ ​

Select answer

Drag and Drop

The graph of a linear function is shown on the grid.

The equation that is best represented by this graph is:



y =

Placeholder

x + ​

Placeholder

Drag these tiles and drop them in the correct blank above

7

4

Multiple Choice

The table represents some points on the graph of linear function f.

Which function represents f?

• f(x) = 26(x - 2)

• f(x) = -26(2x - 1)

• f(x) = 13(x - 2)

• f(x) = -2(26x - 1)

Multiple Choice

Researchers in Antarctica discovered a warm sea current under a glacier that is causing the glacier to melt. The ice shelf of the glacier had a thickness of approximately 450 m when it was first discovered. The thickness of the ice shelf is decreasing at an average rate of 0.06 m per day.

Which function can be used to find the thickness of the ice shelf in meters x days since the discover?

• t(x) = 450 - 0.06x

• t(x) = -0.06(x + 450)

• t(x) = 450 + 0.06x

• t(x) = -0.06(x + 450)

Drag and Drop

The table shows the amount of pet food in cups remaining in an automatic feeder as a function of the number of meals the feeder has dispensed.

Based on the table, the function that best models this situation is:



f(n) = ​

Placeholder

n + ​

Placeholder

Drag these tiles and drop them in the correct blank above

-3

24

16

8

64

Multiple Choice

The conversion of degrees Celsius to degrees Fahrenheit can be represented by a linear relationship. The graph shows the linear relationship between y, the temperature in degrees Fahrenheit, and x, the temperature in degrees Celsius from the freezing point of water.

Which equation best represents this situation?

• $$y=\frac{5}{9}x$$

• $$y=\frac{5}{9}x+32$$

• $$y=\frac{9}{5}x$$

• $$y=\frac{9}{5}x+32$$

Multiple Choice

At a restaurant, jars of tomato sauce are stored in boxes in the pantry. Each box contains 8 jars of tomato sauce. A cook uses 2 jars from 1 of the boxes.

Which function shows the relationship between y, the total number of jars of tomato sauce remaining in the pantry, and x, the number of boxes in the pantry?

• y = 6x

• y = 8x

• y = 8x - 2

• y = 8x - 1

Multiple Choice

The graph of a linear function is shown on the grid.

Which equation is best represented by this graph?

BE CAREFUL - HINT: check by graphing the equation in Desmos to make sure it matches the given graph.

• $$y+2=\frac{7}{5}\left(x+7\right)$$

• $$y-2=\frac{7}{5}\left(x-7\right)$$

• $$y+2=\frac{5}{7}\left(x+7\right)$$

• $$y-2=\frac{5}{7}\left(x-7\right)$$