Question

Given the system of linear equations: [A]{x} = {C}, where the coefficient matrix [A] is decomposed to: U = [1 2; 0 -9 3; 0 0 -15], find the solution for {x} using backward substitution.

Answer 1

Backward substitution is a method used to solve a system of linear equations when the coefficient matrix is an upper triangular matrix. It involves solving each equation for one variable, starting from the last equation. The solution for {x} can be found by substituting the values of each variable into the previous equations.

Step-by-step explanation:

Backward substitution is a method used to solve a system of linear equations when the coefficient matrix is an upper triangular matrix. To use backward substitution, start by solving the last equation for the last variable. Then substitute the value of that variable into the second-to-last equation and solve for the second-to-last variable. Repeat this process until all variables have been solved for. In this case, the coefficient matrix [A] is given as U = [1 2; 0 -9 3; 0 0 -15], so we can use backward substitution to solve for the variables in the equation [A]{x} = {C}.

Step 1: Solve the last equation for the last variable:

-15x3 = c3

x3 = c3/-15

Step 2: Substitute x3 into the second-to-last equation and solve for x2:

-9x2 + 3x3 = c2

-9x2 + 3(c3/-15) = c2

Step 3: Substitute x3 and x2 into the first equation and solve for x1:

x1 + 2x2 = c1

x1 + 2(-9x2 + 3(c3/-15)) = c1

This gives us the values of x1, x2, and x3 that satisfy the equation [A]{x} = {C}.