Example 1 My favorite two-digit number has the following properties: if you add its digits together, you get 6. If you switch its digits, you get a number that’s 18 less than the original number. What is my number?

Solution Let’s call my number “xy”, so that its first digit is x and it second digit is y. The problem tells us x + y = 6. Since x is in the tens place and y is in the ones place, my number is equal to 10x + y. If we switch the digits, this new number is equal to 10y + x. We know that the original number is 18 greater than the new number, so we can write 10x + y = 10y + x Now we can solve this system by substitution.

Solution (cont.) The first equation tells us that x = 6 – y. We can substitute for x in the second equation. 10(6 – y) + y = 10y + (6 – y) – 9y = 9y y = 36 y = 2 Plugging this back into the first equation, x = 4. Thus, my number is 42.

Graph Graphing our system confirms our answer: the two lines intersect at the point (4, 2).

Example 2 I sold a total of 300 ice cream cones and made $1200 today at my ice cream store. If large ice cream cones cones cost $5 and small ice cream cones cost $2, how many of each type of cone did I sell?

Solution If x is the number of small cones we sell and y is the number of large cones, we can form two equations. First, x + y = 300, since we sold a total of 300 cones. Second, 2x + 5y = 1200, since we get $2 for each small cone and $5 for each large cone, and the total must be $1200. We can solve this system using substitution.

Solution (cont.) x + y = 300, so y = 300 – x. Plugging this into the second equation, we get: 2x + 5(300 – x) = x = x = -300 x = 100 Plugging this into the first equation, we see that y equals 200. We sold a total of 100 small cones and 200 large cones.

Graph Again, graphing our equations confirms our answer.

Example 3 Say there are two high-speed internet providers in your city: Company A and Company B. Company A charges $125 per year, plus another $1 per gigabyte you download. Company B charges $25 per year, plus another $3 per gigabyte you download. How much do you have to download for Company A’s offer to be better?

Solution We can express the cost of each company’s plan as a function of the amount of data you download. Call y the cost of each plan and call x the amount of data you download, in gigabytes. Company A’s plan follows the equation y = x. Company B’s plan follows the equation y = x. The two plans will be equally expensive when both equations are satisfied. We can solve the system by substitution.

Solution (cont.) Substitute x for y in the first equation. This gives us x = x. 2x = 100 x = 50 If you download more than 50 gigabytes in a year, company A’s plan will be the better deal.

Graph We can check our answer using a graph.