Find values of x for which 3xx1=3421.
Get the complete, step-by-step math solution for: "Find values of x for which {ll}3 & x \\ x & 1 = {ll}3 & 2 \\ 4 & 1 .". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Calculate the determinant of the left-hand side matrix
To find the determinant of a 2×2 matrix (acbd), we use the formula ad - bc. Applying this to the left-hand side matrix, we multiply the elements on the main diagonal (3×1) and subtract the product of the elements on the anti-diagonal (x×x).
Step 2: Calculate the determinant of the right-hand side matrix
Similarly, for the right-hand side matrix, we apply the same determinant formula. We multiply 3×1 and subtract 2×4, which simplifies to 3−8, resulting in −5.
Step 3: Equate the determinants and solve for x
Now we set the two calculated determinants equal to each other, as given in the problem statement. This forms an algebraic equation that we can solve for x.
Step 4: Isolate x squared
To solve for x, we first isolate the x2 term. We subtract 3 from both sides of the equation, which gives us −x2=−8. Then, we multiply both sides by −1 to get x2=8.
Step 5: Find the values of x
Finally, to find x, we take the square root of both sides of the equation x2=8. Remember that taking the square root yields both positive and negative solutions. We can simplify 8 as 4×2, which is 22.