Transcript 3x y = 6
Solving Linear Systems Algebraically Quiz 1. Solve by graphing: 3x + y = -6 y = 2x – 1 3x + y = –6 y = 2x – 1 You must solve the first equation for y before you can graph it 3x + y = -6 -3x -3x y = 6 – 3x Solution (–1, –3) Solve systems, substitution Systems of equations can be solved algebraically rather than graphically. One method is called substitution. Example: 2x + 3y = 8 and x + 4y = 9 We are going to solve one of the equations for either x or y – we must decide which will be easier: •The easiest thing to solve for is a variable without a coefficient (such as x in the 2nd equation •If there is not a variable without a coefficient look for a coefficient that will divide into the other numbers in the equation without leaving a fraction Take the piece in We will solve the 2nd equation for x: the circle and 2x + 3y = 8 x + 4y = 9 - 4y -4y x = 9 – 4y Now that we have 2(9 - 4y) + 3y = 8 solved it for x we will We will now solve this use that in the first equation for y equation to solve for y substitute it in for x in the equation putting it in parenthesis Example (cont): So far we have the original 2 equations, the equation we solved for x, and the piece that we substituted in: 2x + 3y = 8 x + 4y = 9 x = 9 – 4y 2(9 - 4y) + 3y = 8 We now need to solve this equation for y 18 – 8y + 3y = 8 Distribute the 2 We will plug 2 in for y in either of the 18 – 5y = 8 Combine -8y and 3y original equations: - 18 -18 Subtract 18 from both sides x + 4y = 9 -5y = -10 Divide both sides by -5 x + 4(2) = 9 -5 -5 x+8=9 y = 2 We will use this value to find x - 8 -8 So our solution to this problem is: x=1 Solution (1, 2) Remember that the solution to a system is the point that the cross, so (1,2) would be the point these two lines cross at if they were graphed Example: Solve 2x + 4y = 14 and 2y – 3x = -5 There is no variable without a coefficient so we look at the coefficients and try to pick one out that will divide into the other coefficients. In the 1st equation 2 will divide into 4 and 14 without leaving any fractions so we solve for x. 2x + 4y = 14 - 4y -4y 2x = 14 – 4y 2 2 2 x = 7 – 2y 2y – 3x = -5 2y – 3(7 – 2y) = -5 2y – 21 + 6y = -5 8y – 21 = -5 + 21 +21 Now we plug this in for 8y = 16 x in the 2nd equation 8 8 y=2 We now plug this into one of the equations 2x + 4y = 14 2x + 4(2) = 14 2x + 8 = 14 -8 -8 2x = 6 2 2 x=3 Solution – (3,2) How do we check a solution that we are given or that we find? Simply plug the alleged solution into both equations – if it works in both equations then it is a solution…if it doesn’t work in either or both, then it isn’t a solution. Is (-2,3) a solution for the following system: Plug (-2,3) into each equation and see if it 2x – 3y = -13 works -3x + 4y = 18 2x – 3y = -13 2(-2) – 3(3) = -13 -4 – 9 = -13 -13 = -13 (true) -3x + 4y = 18 -3(-2) + 4(3) = 18 6 +12 = 18 18 = 18 (true) Since (2,3) works in both equations it is the solution to the system Find the solution to the following system using the substitution method and check your answer: 2x – y = 5 4x + 2y = 22 Homework for tonight is 7.2 Remember that the Chapter 5 Extra Credit is due next Tuesday (the 9th)