System of Equations There are 3 types of systems, and 3 ways to solve them, see this next example. One Possible Solution The Most Common Vacation Jamie and Tia are saving money so they can take a trip after graduation. Jamie already has $650 in the bank, and saves $15 a week. Tia saves $25 a week, and already has $180 in the bank. How many weeks will it take for both girls to have the same amount of money in the bank, and how much will that be? Example #1 If the amount of money is represented by "x", and if the weeks was represented by "y," the system could look like this: { y = 25x + 180

y = 15x + 650 Since the "y" value is already defined, the best method is substitution. y = 25x +180 y = 15x + 650 { y = 25x + 180 (25x + 180) = 15x + 650 -180 -180 25x = 15x + 470 -15x -15x 10x = 470 x = 47 10 10 Jamie and Tia will have the same amount of money in 47 weeks. Now to Solve for "y" y = 25x + 180 y = 15x + 650 { y = 25x + 180 y = 25(47) + 180 y = 1175 + 180 y = 1355 When Jamie and Tia have the same amount of money, it will be $1355. Now to Check Our Solution y = 25x + 180 y = 15x + 650 { y = 15x + 650 1355 = 15(47) + 650 1355 = 705 + 650 1355 = 1355 So after 47 weeks (x), Jamie and Tia will each have $1355 (y) in the bank. Example 2 A Special Case No Possible Solutions It's also possible for there to be a system of equations with no solution, see the next system for an example. Two tubs are filled with water. Tub A is filled with 23 gallons of water, Tub B is filled with 19 gallons of water. Each tub drains 2 gallons of water each minute. How long will it take for the tubs to have the same amount of water, and how much would that be? Water Level If the amount of water in the tub is represented by "y," and if the amount of time (in minutes) is represented by "x," the system could look like this: y = -2x + 23 4x = -2y + 38 { Let's try solving this equation by graphing, but first we will need to rewrite the 2nd equation. Let's rewrite this equation into slope intercept form. 4x = -2y + 38 -4x -4x 0 = -4x - 2y + 38 +2y +2y 2y = -4x + 38 Write the Original Equation Subtract 4x from each side Add 2y to each side 2 2 y = -2x + 19 Substitute for "y" Subtract 180 from both sides Subtract 15x from both sides Divide each side by 10 First let's Substitute to find the value of "x" x = 47 Multiply Add x = 47, y = 1355 Multiply Add Now both equations are in slope intercept form, now we are ready to graph! 0 5 10 15 20 25 Water (In gallons) Time (In minutes) 5 10 15 20 25 Water Levels First let's look at the equation for graphing a straight line. y = mx + b m is the slope of the equation

b is the y intercept y = -2x + 19 19 is the y-intercept, that is the "y" value of the ordered pair the line passes through when "x" is equal to 0. In this case, the y-intercept is the ordered pair (0, 19) The number marked in green is the slope, the slope is a ratio that tells how the line is slanted. The ratio is Rise (y) over Run (x) -2 Rise 1 Run This pattern would go on forever, so let's skip ahead to when both lines are graphed. Since both lines have the same slope, they will never intersect, meaning, this equation has no possible solutions. Please watch in full screen for full effect. Is it possible to have infinite solutions? Check this next example. Example 3 A Special Case Infinite Solutions School "x" has the same amount of students as school "y." School has 867 students. How many students does each school have? Students If "x" represented the amount of students in the first school, and "y" represented the number of students in the 2nd school, the system could look like this: { y = x x - y = 0 We should be able to Eliminate to find the value of "x." y = x x - y = 0 { - y = -x x - y = 0 + Eliminate "y" x - 2y = -x First, let's Eliminate to find the value of "x" School x has 994 students. We already have the value for school y, so let's find the value for "x" y(-1) = x(-1) +x +x 2x -2y = 0 +2y +2y 2x = 2y x = y In this equation, x is equal to y, so there is an infinite number of solutions, as long as x is the same value is y.

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