Exercise: Find the Ratio x : y

Here are the solutions for finding the ratio x:y for each problem. 1. x^2 - 4xy + 3y^2 = 0 Step 1: Divide the entire equation by y^2 (assuming y ≠ 0). (x^2)/(y^2) - (4xy)/(y^2) + (3y^2)/(y^2) = 0 ((x)/(y))^2 - 4((x)/(y)) + 3 = 0 Step 2: Let k = (x)/(y). Substitute k into the equation and solve the quadratic. k^2 - 4k + 3 = 0 (k-1)(k-3) = 0 Step 3: Solve for k. k-1=0 k=1 k-3=0 k=3 Step 4: Substitute back k = (x)/(y) and express as a ratio. (x)/(y) = 1 or (x)/(y) = 3 The ratio x:y is 1:1 or 3:1. 2. 2x^2 - 5xy + 2y^2 = 0 Step 1: Divide the entire equation by y^2 (assuming y ≠ 0). 2((x)/(y))^2 - 5((x)/(y)) + 2 = 0 Step 2: Let k = (x)/(y). Substitute k into the equation and solve the quadratic. 2k^2 - 5k + 2 = 0 (2k-1)(k-2) = 0 Step 3: Solve for k. 2k-1=0 k=(1)/(2) k-2=0 k=2 Step 4: Substitute back k = (x)/(y) and express as a ratio. (x)/(y) = (1)/(2) or (x)/(y) = 2 The ratio x:y is 1:2 or 2:1. 3. 3x^2 + xy - 2y^2 = 0 Step 1: Divide the entire equation by y^2 (assuming y ≠ 0). 3((x)/(y))^2 + ((x)/(y)) - 2 = 0 Step 2: Let k = (x)/(y). Substitute k into the equation and solve the quadratic. 3k^2 + k - 2 = 0 (3k-2)(k+1) = 0 Step 3: Solve for k. 3k-2=0 k=(2)/(3) k+1=0 k=-1 Step 4: Substitute back k = (x)/(y) and express as a ratio. (x)/(y) = (2)/(3) or (x)/(y) = -1 The ratio x:y is 2:3 or -1:1. 4. (x+3y)/(x) = (7)/(4) Step 1: Separate the fraction on the left side (assuming x ≠ 0). (x)/(x) + (3y)/(x) = (7)/(4) 1 + 3(y)/(x) = (7)/(4) Step 2: Isolate (y)/(x). 3(y)/(x) = (7)/(4) - 1 3(y)/(x) = (7)/(4) - (4)/(4) 3(y)/(x) = (3)/(4) Step 3: Solve for (y)/(x) and then find (x)/(y). (y)/(x) = (1)/(4) (x)/(y) = 4 Step 4: Express as a ratio. The ratio x:y is 4:1. 5. (2x-y)/(y) = 5 Step 1: Separate the fraction on the left side (assuming y ≠ 0). (2x)/(y) - (y)/(y) = 5 2(x)/(y) - 1 = 5 Step 2: Isolate (x)/(y). 2(x)/(y) = 5 + 1 2(x)/(y) = 6 Step 3: Solve for (x)/(y). (x)/(y) = 3 Step 4: Express as a ratio. The ratio x:y is 3:1. 6. (3x)/(y) + (8y)/(x) = 10 Step 1: Let k = (x)/(y). Then (y)/(x) = (1)/(k). Substitute these into the equation (assuming x, y ≠ 0). 3k + (8)/(k) = 10 Step 2: Multiply the entire equation by k to eliminate the denominator. 3k^2 + 8 = 10k Step 3: Rearrange into a quadratic equation and solve for k. 3k^2 - 10k + 8 = 0 (3k-4)(k-2) = 0 Step 4: Solve for k. 3k-4=0 k=(4)/(3) k-2=0 k=2 Step 5: Substitute back k = (x)/(y) and express as a ratio. (x)/(y) = (4)/(3) or (x)/(y) = 2 The ratio x:y is 4:3 or 2:1. 7. 4x^2 - 12xy + 9y^2 = 0 Step 1: Recognize the equation as a perfect square trinomial. (2x)^2 - 2(2x)(3y) + (3y)^2 = 0 (2x - 3y)^2 = 0 Step 2: Take the square root of both sides. 2x - 3y = 0 Step 3: Isolate (x)/(y). 2x = 3y (x)/(y) = (3)/(2) Step 4: Express as a ratio. The ratio x:y is 3:2. 8. (5x+y)/(2x) = 2 Step 1: Multiply both sides by 2x (assuming x ≠ 0). 5x+y = 2(2x) 5x+y = 4x Step 2: Isolate x and y terms. y = 4x - 5x y = -x Step 3: Solve for (x)/(y). (x)/(y) = -1 Step 4: Express as a ratio. The ratio x:y is -1:1. 9. (x)/(2y) + (12y)/(x) = 7 Step 1: Let k = (x)/(y). Then (x)/(2y) = (1)/(2)k and (12y)/(x) = (12)/(k). Substitute these into the equation (assuming x, y ≠ 0). (1)/(2)k + (12)/(k) = 7 Step 2: Multiply the entire equation by 2k to eliminate the denominators. k^2 + 24 = 14k Step 3: Rearrange into a quadratic equation and solve for k. k^2 - 14k + 24 = 0 (k-2)(k-12) = 0 Step 4: Solve for k. k-2=0 k=2 k-12=0 k=12 Step 5: Substitute back k = (x)/(y) and express as a ratio. (x)/(y) = 2 or (x)/(y) = 12 The ratio x:y is 2:1 or 12:1. 10. 5x^2 - 13xy + 6y^2 = 0 Step 1: Divide the entire equation by y^2 (assuming y ≠ 0). 5((x)/(y))^2 - 13((x)/(y)) + 6 = 0 Step 2: Let k = (x)/(y). Substitute k into the equation and solve the quadratic. 5k^2 - 13k + 6 = 0 (5k-3)(k-2) = 0 Step 3: Solve for k. 5k-3=0 k=(3)/(5) k-2=0 k=2 Step 4: Substitute back k = (x)/(y) and express as a ratio. (x)/(y) = (3)/(5) or (x)/(y) = 2 The ratio x:y is 3:5 or 2:1. Send me the next one 📸