Final answer:
To solve the given simultaneous equations in matrix form, we first express them in matrix notation and then multiply the inverse of the coefficient matrix by the constants matrix to find the values of y₁ and y₂, which are 2 and 4 respectively.
Step-by-step explanation:
To solve the simultaneous equations 5y₁ + 2y₂ = 8 and 3y₁ + 7y₂ = 28 using matrix algebra, we follow these steps:
Matrix Notation
Firstly, we write these equations in matrix form (A):
[[5, 2], [3, 7]] * [y₁, y₂] = [8, 28]. So the coefficient matrix is A = [[5, 2], [3, 7]], the variable matrix is y = [y₁, y₂], and the constant matrix is b = [8, 28].
Solving for y₁ and y₂
Then we calculate the inverse of matrix A and multiply it by matrix b to find matrix y. Matrix A inverse is calculated as A⁻¹ = 1/(ad-bc) * [[d, -b], [-c, a]], where a, b, c, and d are the elements of matrix A. In this case, A⁻¹ = 1/(5*7-3*2) * [[7, -2], [-3, 5]] = 1/29 * [[7, -2], [-3, 5]]. Multiplying A⁻¹ by b gives us the solution y = A⁻¹ * b. After the multiplication, we find y₁ = 2 and y₂ = 4.