Question

3x-y=1 x^2-y=-1Решить систему уравнения ​

Answer 1

3x - y = 1

x² - y = -1

Вычтем из второго первое

х² - 3х = -2

х² - 3х + 2 = 0

(х - 1)(х - 2) = 0

х₁ = 1, у₁ = х₁² + 1 = 2

х₂ = 2,  у₂ = х₂² + 1 = 5

Answer 2

To solve the system of equations, we can use the method of substitution or elimination. Let's use the substitution method:

Step 1: Solve one of the equations for one variable in terms of the other variable. Step 2: Substitute the expression obtained in Step 1 into the other equation. Step 3: Solve the resulting equation for the remaining variable. Step 4: Substitute the value found in Step 3 back into either of the original equations to find the value of the other variable.

Given equations:

1. 3x - y = 1 ......(i)
2. x^2 - y = -1 ......(ii)

Step 1: Solve equation (i) for y in terms of x: 3x - y = 1 y = 3x - 1

Step 2: Substitute the expression for y from Step 1 into equation (ii): x^2 - (3x - 1) = -1

Step 3: Solve for x: x^2 - 3x + 1 = -1 x^2 - 3x + 2 = 0

Now, let's solve the quadratic equation for x. Factoring doesn't work here, so we'll use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a

where a = 1, b = -3, and c = 2

x = (3 ± √((-3)^2 - 4 * 1 * 2)) / 2 * 1 x = (3 ± √(9 - 8)) / 2 x = (3 ± √1) / 2 x = (3 ± 1) / 2

Now, we have two possible solutions for x: a) x = (3 + 1) / 2 = 4 / 2 = 2 b) x = (3 - 1) / 2 = 2 / 2 = 1

Step 4: Substitute the values of x back into the equation we obtained in Step 1 to find y:

For x = 2: y = 3x - 1 = 3 * 2 - 1 = 6 - 1 = 5

For x = 1: y = 3x - 1 = 3 * 1 - 1 = 3 - 1 = 2

So, the solutions to the system of equations are: x = 2, y = 5 x = 1, y = 2