Solving Problems with Systems of Equations

Warmup

Replace the question marks by entering the coordinates of where the lines meet and press the Enter key. Then click “New Problem” for another system of equations.

Activity #1

Solving Problems with Systems of Equations.

• Solve each problem below. Explain or show your reasoning.

(1.) Two friends live 7 miles apart. One Saturday, the two friends set out on their bikes at 8 am and started riding towards each other. One rides at 0.2 miles per minute, and the other rides at 0.15 miles per minute. At what time will the two friends meet?

(2.) Students are selling grapefruits and nuts for a fundraiser. The grapefruits cost $1 each and a bag of nuts cost $10 each. They sold 100 items and made $307. How many grapefruits did they sell?

(3.) Jada earns $7 per hour mowing her neighbors’ lawns. Andre gets paid $5 per hour for the first hour of babysitting and $8 per hour for any additional hours he babysits. What is the number of hours they both can work so that they get paid the same amount?

(4.) Invent another problem that is like one of these, but with different numbers.

(5.) Solve your problem.

Activity #2

Solving Linear System Equations Graphically.

In the applet below, note the system of equations displayed. One equation is displayed in pink. The other equation is displayed in purple.

•  Move the pink points (of the pink line) so that the line becomes the graph of the pink equation displayed. (You’ll get confirmation once you’ve done this correctly.) ​
•  Move the purple points (of the purple line) so that the line becomes the graph of the purple equation displayed. (You’ll also get confirmation once you’ve done this correctly.) ​
•  Enter the respective x– and y- coordinates of the solution to this system of equations. If you enter this correctly, the applet will confirm this. 
• Click . Applet generates problems for which coordinates of solutions are integers.
• Repeat this process as many times as you need in order to master this concept! 

Activity #3

Solve Linear Systems Algebraically.

Solve the pair of equations below by any method.

• Enter your values in the boxes.
• Click .
• Repeat this process as many times as you can in order to master this concept! 

Challenge #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.

Challenge #2

Consecutive numbers follow one right after the other. An example of three consecutive numbers is 17, 18, and 19. Another example is -100, -99, -98.

 How many sets of two or more consecutive positive integers can be added to obtain a sum of 100?

Challenge #3

A car is driving towards home at 0.5 miles per minute. If the car is 4 miles from home at t = 0, which of the following can represent the distance that the car has left to drive?