Question

Решите системы уравнения способом сложения или способом подстановки: a) 5x-2y=11 б)3x+5y=12 4x-y=4 x-2y=-7

Answer 1

Первая - сложением, вторая - подстановкой

Answer 2

a) To solve the system of equations using the method of addition (elimination method):

1. 5x - 2y = 11
2. 3x + 5y = 12

Step 1: Multiply both sides of equation 1 by 5 and equation 2 by 2 so that the coefficients of y will cancel when we add the equations. (5) * (5x - 2y) = (5) * 11 (2) * (3x + 5y) = (2) * 12

This gives us: 3. 25x - 10y = 55

4. 6x + 10y = 24

Step 2: Add equations 3 and 4 together to eliminate y. (25x - 10y) + (6x + 10y) = 55 + 24

This simplifies to: 31x = 79

Step 3: Solve for x. x = 79 / 31

Now, find the value of y by substituting the value of x into either equation 1 or 2: 5x - 2y = 11 5 * (79 / 31) - 2y = 11 395 / 31 - 2y = 11

Step 4: Solve for y. 2y = 395 / 31 - 11 2y = (395 - 341) / 31 2y = 54 / 31

y = (54 / 31) / 2 y = 54 / 62 y = 27 / 31

So, the solution to the system of equations is: x ≈ 2.548 y ≈ 0.871

b) To solve the system of equations using the method of substitution:

1. 4x - y = 4
2. x - 2y = -7

Step 1: Solve one of the equations for one variable in terms of the other variable. Let's solve equation 1 for y: y = 4x - 4

Step 2: Substitute the expression for y from step 1 into equation 2: x - 2(4x - 4) = -7

Step 3: Simplify and solve for x: x - 8x + 8 = -7 -7x + 8 = -7 -7x = -15

x = -15 / -7 x ≈ 2.143

Step 4: Substitute the value of x into the expression for y from step 1: y = 4(2.143) - 4 y ≈ 8.572 - 4 y ≈ 4.572

So, the solution to the system of equations is: x ≈ 2.143 y ≈ 4.572