Examples from your Text: Page 212 # 30) 𝑦 = −2 { 2𝑥 + 3𝑦 = 4 This one couldn’t be easier. I already know what y is. Just plug in to find x 2x + 3(-2) = 4 2x – 6 = 4 2x = 10 X=5 so my solution is (5,-2) This is the point where the two lines intersect, and the only pair of (x,y) that will make both equations true. # 42) { 3𝑥 + 7𝑦 = −5 𝑦 = 6𝑥 − 5 Since the y is already isolated in the second equation we simply plug 6x – 5 into the first equations “y” 3x + 7(6x – 5) = -5 3x + 42x – 35 = -5 45x – 35 = -5 45x = 30 X = 30/45 or x = 2/3 or x = .66667 Now we plug this into any equation. I choose the 2 nd equation. Y = 6(.66667) – 5 Y=4–5 Y = -1 So our solution is (2/3, -1) This is the point where the two lines intersect, and the only pair of (x,y) that will make both equations true. 6𝑥 + 2𝑦 = 7 # 54) { 𝑦 = −3𝑥 + 2 Plug -3x + 2 into the y in the first equation 6x + 2(-3x + 2) = 7 6x -6x + 4 = 7 4=7 SPECIAL CASE: This can’t be true! This system is called INCONSISTENT! What that means is that there is no solution, in other words there is no x and y we can choose that will work for both equations. Visually it means that the lines never touch (parallel). 3𝑥 − 12𝑦 = −24 # 56) { −𝑥 + 4𝑦 = 8 Since no variable is isolated we will do it ourselves. I choose to isolate the x in equation 2 (because it looks easiest to do) EQ2: -x + 4y = 8 -x = -4y + 8 x = 4y – 8 Now I will replace the x in equation one with 4y – 8 3(4y – 8) -12y = -24 12y -24 – 12y = -24 -24 = -24 SPECIAL CASE: This is always true. These are the same line! So how do we express it? Your solution will be written like it’s a point, kinda. Start out with (x, …) This indicates x could be anything…but once we pick x, what is y. Look at equation 2 again.. -x + 4y = 8 Lets find the y 4y = x + 8 y = .25x + 2 So our solution is (x, .25x + 2) This describes all the possible points where the lines touch.