Final answer:
To solve the matrix equation [[2,5],[1,3]][[x],[y]]=[[7],[5]], we can write it as (2x + 5y = 7) and (x + 3y = 5). Here are the step-by-step instructions to solve for x and y.
Step-by-step explanation:
To solve the matrix equation [[2,5],[1,3]][[x],[y]]=[[7],[5]], we can write it as (2x + 5y = 7) and (x + 3y = 5). We can solve this system of equations using matrix operations. Here is the step-by-step solution:
- Write the augmented matrix. [[2,5,7],[1,3,5]]
- Apply row operations to transform the augmented matrix into row-echelon form or reduced row-echelon form. In this case, we can subtract the first row multiplied by 1/2 from the second row.
- The resulting matrix becomes [[2,5,7],[0,1/2,-1/2]].
- From the matrix, we can deduce the equations: 2x + 5y = 7 and (1/2)y = -1/2.
- Solve the second equation: y = -1.
- Substitute the value of y into the first equation to solve for x: 2x + 5(-1) = 7. Simplifying, 2x - 5 = 7, 2x = 12, and x = 6.
- The solution to the matrix equation is x = 6 and y = -1.