Systems of Equations


Setting up linear equations.

  • In the input box on the left type f(x) = mx+b and press the Enter key on your keyboard.
  • Check the radio buttons(circles) on the left. Two sliders for variables m and b should appear on the graph.​
  • Move each slider to check that it is working. ​To delete any unwanted features, click the 3 vertical dots
  • Then answer the questions that follow.

(1). What do you notice when you change the variable m ? What type of transformation happens?

(2). What do you notice when you change the variable b ? What type of transformation happens?

(3). What is the parent function of the linear equation?

(4). How does the end behavior change as the slope changes?

Now, enter the following functions; , ,

You can always recall each graph by checking and unchecking the radio button on the left.

Use the sliders for m and b to answers questions 5-14.

(5). How can f(x) be parallel to g(x)?

(6). Did you have to change m, b, or both? Why?

(7). How can f(x) intersect h(x)? Does both lines always intersect?

(8). If f(x) and h(x) do not intersect what do you notice?

(9). How can f(x) be perpendicular to p(x)?

(10). What is the relationship called for slopes that are perpendicular to each other?

(11). What is the rise and run of the slope m for f(x)?

(12). What is true when negative reciprocals are multiplied together?

(13). Does b have any effect making lines perpendicular? Why?

(14). Can you make f(x) and p(x) the same line? How many points do they share?

(15). How is it similar to parallel lines?


  • Click the Tools icon in the blue tab above the input box (the overlapping triangle and circle).  ​
  • Click point. Then click the intersection of f(x) and p(x) to make point a. ​
  • Click on line f(x) to make point b. Click on line p(x) to make point c. ​
  • Click more under the tools section. Scroll to the section that says “Measure.” ​
  • Click on Angle icon. Then Click on points c, a, and b in that order. ​
  • It should tell you the degree of the angle and you can see if if it is parallel or not.

(16). In the input box create two more linear equations and make f(x) parallel and perpendicular one of each. Take a screenshot of your answers and submit it online.


Draw and Interpret a Straight Line Graph.

There is a hiking trail near the town where Han and Jada live that starts at a parking lot and ends at a lake. Han and Jada both decide to hike from the parking lot to the lake and back, but they start their hikes at different times.

At the time that Han reaches the lake and starts to turn back, Jada is 0.6 miles away from the parking lot and hiking at a constant speed of 3.2 miles per hour toward the lake. Han’s distance, d from the parking lot can be expressed as d = -2.4 + 4.8, where  represents the time in hours since he left the lake.

(1). What is an equation for Jada’s distance from the parking lot as she heads toward the lake?

(2). Draw both graphs: one representing Han’s equation and one representing Jada’s equation. It is important to be very precise.

(3). Find the point where the two graphs intersect each other. What are the coordinates of this point?

(4). What do the coordinates mean in this situation?

(5). What has to be true about the relationship between these coordinates and Jada’s equation?

(6). What has to be true about the relationship between these coordinates and Han’s equation?


Write an Equation from a Graph.

Here is the graph for one equation in a system of equations:

(1). Write a second equation for the system so it has infinitely many solutions.

(2). Write a second equation whose graph goes through (0, 1) so the system has no solutions.

(3). Write a second equation whose graph goes through (0, 2) so the system has one solution at (4, 1).


Here is an equation that is true for all values of  :  5(+ 2) = 5+ 10. Elena saw this equation and says she can tell 20(+ 2) + 31 = 4(5+ 10) + 31  is also true for any value of . How can she tell? Explain your reasoning.


The temperature in degrees Fahrenheit, , is related to the temperature in degrees Celsius, , by the equation

(1). In the Sahara desert, temperatures often reach 50 degrees Celsius. How many degrees Fahrenheit is this?

(2). In parts of Alaska, the temperatures can reach -60 degrees Fahrenheit. How many degrees Celsius is this?

(3). There is one temperature where the degrees Fahrenheit and degrees Celsius are the same, so that = . Use the expression from the equation, where is expressed in terms of , to solve for this temperature.


A stack of n small cups has a height,  h, in centimeters where h = 1.5n + 6. A stack of n large cups has a height, h, in centimeters where h = 1.5n + 9.

(1). Graph the equations for each cup on the same set of axes. Make sure to label the axes and decide on an appropriate scale.

(2). For what number of cups will the two stacks have the same height?