Question

40б решить уравнение всеми ТРЕМЯ способами x-y=-1 и 2x+y=4

Answer 1

Ответ в приложении снизу:

Дополню

Answer 2

To solve the system of equations x - y = -1 and 2x + y = 4, we can use three different methods: the substitution method, the elimination method, and the matrix method. Let's go through each of them step by step:

Method 1: Substitution Method

Step 1: Solve one equation for one variable in terms of the other variable. x - y = -1 → x = y - 1

Step 2: Substitute the expression for x from Step 1 into the other equation and solve for y. 2(y - 1) + y = 4 2y - 2 + y = 4 3y - 2 = 4 3y = 6 y = 2

Step 3: Substitute the value of y back into one of the original equations and solve for x. x = 2 - 1 x = 1

So, the solution to the system of equations is x = 1 and y = 2.

Method 2: Elimination Method

Step 1: Multiply the first equation by 2 to make the coefficients of y in both equations equal. 2(x - y) = 2(-1) → 2x - 2y = -2

Step 2: Add the modified first equation to the second equation to eliminate the y variable. (2x - 2y) + (2x + y) = -2 + 4 4x - y = 2

Step 3: Solve the resulting equation for one variable. 4x - y = 2 → y = 4x - 2

Step 4: Substitute the expression for y from Step 3 into one of the original equations and solve for x. x - (4x - 2) = -1 x - 4x + 2 = -1 -3x + 2 = -1 -3x = -3 x = 1

Step 5: Substitute the value of x back into the expression for y from Step 3 and solve for y. y = 4(1) - 2 y = 4 - 2 y = 2

So, the solution to the system of equations is x = 1 and y = 2.

Method 3: Matrix Method

Step 1: Write the system of equations in matrix form (AX = B). | 1 -1 | | x | | -1 | | 2 1 | x | y | = | 4 |

Step 2: Take the inverse of the coefficient matrix A^-1. | 1 -1 |^-1 = | 1 1 | | 2 1 | | -2 1 |

Step 3: Multiply both sides by A^-1. | x | | 1 1 | | -1 | | y | = | -2 1 | x | 4 |

Step 4: Solve for x and y. x = (1 * -1) + (1 * 4) = -1 + 4 = 3 y = (-2 * -1) + (1 * 4) = 2 + 4 = 6

So, the solution to the system of equations is x = 3 and y = 6.

In all three methods, we found that the solution to the system of equations is x = 1 and y = 2.